- Waves encountering a turning point (4.3 MB gif file): To create this movie a background density field was used for which the buoyancy frequency decreases with height. Waves are generated using an oscillatory forcing term in the vertical momentum equation. The forcing is located at z=0.2 with the bottom and surface of the fluid being at z = 0 and 1 respectively. Upward propagating energy encounters a turning point at z=0.5. As the turning point is approached the energy beam turns to become vertical. Energy reflects at the turning point and propagates downward. In these animations the horizontal velocity is shown. Green/purple are negative/positive values of u. Red indicates small values of u. Phase propagation is perpendicular to energy propagation with phase and energy propagating in opposite vertical directions.
- Supercritical flow over a bump (1.2 MB gif file): This shows the waves generated by an impulsively started supercritical flow over a bump. Uniform stratification. Upstream flow speed is U=1. Froude number is 1.2.
- Shoaling solitary wave (0.3 MB gif file): This shows the evolution of a shoaling solitary wave. A single solitary wave in deep water of depth 100m shoals over a narrow shelf edge into water 40m deep. Stratification increases exponentially towards the surface. On the shelf a large leading solitary wave with a trapped core is formed. Trailing are several other waves, including large a mode-2 wave. A small reflected wave can also be detected. Contours are at equal density intervals. Surface to bottom density change is 0.01 in the deep water (nondimensionalized by reference density). Simulations were done using 3000 points in the horizontal, 120 in the vertical.
- Shoaling solitary wave (1.7 MB gif file): Magnified view of previous shoaling wave animation. Only upper 60 m and rightmost 5 km are shown. Temporal resolution is increased by factor of four.
- Nonlinear evolution of a tilted thermocline (1.6 MB gif file): This illustrates nonlinear steepening and the formation of solitary waves in a lake with a simple bathymetry consisting of a deep central region of constant depth and two shelves, lyin below the thermocline, at either end of the lake. The initial state consists of a tilted thermocline at rest. There is no flow through the two endwalls. With no external force to maintain its position, the thermocline levels out at the two side walls as predicted by the linear wave equation. After some time the wave front steepens due to nonlinear effects and trains of solitary-like waves are formed. Both mode-one and mode-two waves can be seen. The evolution is complicated by the depth change.
- Lake Opeongo: case 1 (21 MB gif file): This illustrates nonlinear steepening and the formation of solitary waves in a lake using bathymetry taken from an east-west transect in the South Arm of Lake Opeongo which is in Algonquin Provincial Park, Ontario, Canada. The initial state consists of a tilted thermocline with slope 0.0001 with the fluid at rest. Temperatures above and below the thermocline are 20 and 8 degrees C respectively. The upper panel shows the density field. The horizontal velocity field is shown in the bottom panel (blue-green are negative (leftward) values, red-orange-yellow are positive). No-slip boundary conditions with a viscosity of 10e-5 m^2/s and a diffusivity of 10e-7 m^2/s were used. For this 2D simulation fluid is forced to flow over the ridges in the lake. In the lake itself, or in a 3D simulation, fluid could flow around the ridges and the wave formation process could be significantly different.
- Lake Opeongo: case 2 (12 MB gif file): Same as previous case but with a larger initial thermocline slope of 0.0015.
- Tidal flow over the Knight Inlet Sill (0.69 MB gif file): This illustrates the generation of large internal waves downstream of a tall sill. The topography and density stratification are taken from observations made by David Farmer and Larry Armi at the Knight Inlet Sill, BC. A large overturning wave above the downstream side of the sill creates a thick layer of slow moving fluid. Beneath, a thin downslope jet is formed. This mechanism lies behind the generation of the Chinook winds on the eastern side of the Canadian Rockies. The jet is unstable and strong vortices with associated pulsations in the velocity are created.
