Resonant Internal Gravity Waves

Overview

Internal waves are a special kind of wave that can arise in any fluid that is stratified, meaning that the density of the fluid varies with height. The ocean is a classic example of a naturally occurring stratified fluid, being warmer and less salty near the surface, and colder and saltier in the deep sea. The ocean is stably stratified, meaning that a fluid parcel has a strong tendency to remain at its equilibrium height. Whenever a disturbance of any kind causes vertical motion of fluid, the fluid oscillates around its equilibrium height. This oscillatory disturbance propagates through the fluid as an internal wave. These waves are generally not visible from the surface, but are nearly ubiquitous throughout the ocean.

Although they are still not very well characterized, it is widely believed among oceanographers that internal waves are responsible for a significant fraction of the mixing that must exist to maintain the observed ocean circulation.

We recently proposed that internal waves may actually be responsible for shaping the continental slopes (Zhang, King, and Swinney 2008). This is a consequence of the unusual dispersion relation of internal waves, where the angle of propagation of an internal wave depends only on the frequency of the wave and the rate of increase of density with depth.
In other words, the frequency is related only to the direction of the wave vector, not its magnitude, which is typically the case with waves.

Most internal waves in the ocean are generated by the oscillatory flow of tides over the ocean floor. The strongest waves are generated by the M2 tide, creating internal waves of the same frequency (roughly two cycles per day). The dispersion relation tells us that waves with this frequency should propagate at an angle of around 3 degrees from the horizontal. This is very nearly the angle of many of the continental slopes. Continental slopes connect the shallow continental shelf (typical depth of 140 meters) to the deep ocean floor (typical depth of 4 kilometers). One might be inclined to dismiss this as a coincidence, but the natural angle of repose for the sediments on the continental slopes is much higher (around 20degrees). Clearly, as sediments pile up on the slopes, there is a process besides their natural collapse that is stringently limiting the angle of the continental slopes.

Figure 1. Instantaneous velocity fields of the internal waves with the color denoting velocity magnitude. The waves are much stronger when the critical region of the slope is longer (see the bottom diagram). Note the different color scales for the velocity squared.

Previous experiments have shown that internal waves are generated where the ocean floor slopes at a “critical angle,” where this slope matches the angle at which internal waves travel. Our experiments have shown that a larger generating region at the “critical angle” results in stronger internal waves. In one of the extreme lab cases, the velocity of the waves was over ten times that of the velocity of the background tide. We believe that as sediment piles up on the continental shelves and they gradually become steeper, a critical point is reached where the angle of the shelf matches the angle of internal waves generated by the tides. When this occurs, because of the large area at the critical angle, resonant internal waves of very large amplitude are generated along the continental slope. These intense waves, which we have found to be confined to a very thin boundary layer, keep further material from sedimenting on the slope, thus preserving
the critical angle for the continental slope

It is well established that intense internal waves are generated at specific locations on underwater topography where the slope of the topography matches the angle of propagation of the internal waves. We thought that larger generation regions might lead to stronger waves, and wanted to investigate this phenomena and how pronounced it might be in the oceans, especially along the continental slopes. We thought that these strong waves could provide an explanation for why the angle of the continental slopes is much smaller than the natural angle of repose of sediment.

Setup

All experiments were performed in a 250 liter rectangular glass tank (an aquarium). A model continental slope was mounted on a traverse mechanism and suspended in the tank. The oscillatory motion of the model topography in the lab mimics the oscillating tidal flow in the ocean. Our setup allowed us to vary the oscillation frequency and amplitude.

The fluid was seeded with 10 micron titanium dioxide particles and illuminated with a vertical laser light sheet. Correlation Image Velocimetry analysis enabled us to determine the velocity field many times per second. The stratification was obtained using the ommon “Double Bucket” method (Oster, 1965). Dense saltwater from one tank (the reservoir) is continuously pumped into a second tank initially containing fresh water (the mixing tank). The fluid in the mixing tank gets progressively saltier (and therefore more dense) and is continuously pumped into the bottom of the lab tank. This results in a linearly stratified fluid in the lab tank.

Movies

You can click to watch the movies here (or to save the file, right click and select “Save Link Target as..”):
Raw data from the experiment (21 MB). It is easy to see the intense boundary layer oscillating over the
model topography
Processed Data (12 MB). This movie shows velocity vectors and the color-coding
denotes velocity magnitude.
Dye Resonance (21 MB). A dye layer along a constant-density
line shows the density inversion caused by the internal waves.
denotes velocity magnitude.

Hepeng Zhang, Benjamin King, and Harry L. Swinney “Resonant Generation of Internal Waves on a Model Continental Slope” Physical Review Letters strong>100, 244504 (2008).

final paper: [Journal URL], [pdf], [ps].