Stratified Flows and Internal Waves

Stratified Flows

Many fluid flows of interest consist of a fluid with varying density flowing and interacting under the influence of gravity. The differences in fluid density lead to heavier fluid wanting to ‘sink down’ below lighter less dense fluid. The Earth’s oceans and atmosphere are both stratified, and so such stratified flows are extremely common.

We are particularly interested in producing high and adaptive resolution numerical models of stratified flows. Above right is a picture of the buoyancy (or equivalently density) in a numerical simulation of a gravity current. This is a flow with two regions of constant density (light shown in red, dense shown in purple). The flow starts at the left side of a numerically simulated tank with the dense and light regions separated. This is then released at the beginning of the simulation, and the dense fluid then moves primarily horizontally across the base of the tank as it seeks to find a static equilibrium (the picture shows the flow some time later). The head of the gravity current, the billows and the mixing associated with them are readily apparent.

The second picture on the left shows a similar situation consisting of a fluid with two regions of distinct density, again dense purple fluid and lighter red fluid. This time the initial condition is a vertical separation through the whole height of the tank, at the centre, with dense fluid on the left and light fluid on the right. When the flow is released the fluids ‘exchange’ as they seek a static equilibirum. The intermediate colours again show the mixing between the two fluids that occurs along the interface.

Internal Solitary Waves

Internal waves are waves that occur within a stratified fluid, as opposed to the usual surface waves that we are used to seeing on the ocean. These waves are supported by an underlying stratification of the fluid, and can occur in either the ocean or the atmosphere. An internal solitary wave (ISW) is a form of internal wave that arises due to a balance of nonlinear steepening and dispersive effects. They are large, long waves and can propagate with a steady form for long times.

In our group, we have been particularly interested in studying oceanic ISWs. Thanks to satellite imaging these are now thought to be a ubiquitous feature of the coastal oceans, and may contribute significantly to global ocean mixing. Of particular interest are how such waves interact with their surroundings (both the sea bed beneath them, and the waves at the surface), and how they break. ISWs can break, just as surface waves do, when they move into a shallower region of the ocean. They typically exhibit a shear instability along the crest which then moves downstream on the wave.

We have been involved in both experimental and numerical modelling of ISWs. Experimentally ISWs are produced in a water tank stratified with salt, using a lock-release method. The water in the tank can be seeded with neutrally buoyant particles in order to use particle-image-velocimetry to calculate the velocity field within the wave. Above right is a photograph of a cross-section of such an ISW captured during a breaking event. The image is false-colour, but the bright patches are particles illuminated in the flow (a series of density sensitive probes can also be seen in the upper right corner of the picture). The series of overturning billows can be clearly seen downstream of the peak of the wave on the left. They appear progressively more blurred as the layers of salt and fresh water mix.

Left is a similar picture of a numerical simulation of a breaking ISW. Once again the fluid is stratified with light fluid shown in red and dense in purple. More of the wave can be seen in the frame, and here the front of the wave can be seen clearly. Once again breaking takes the form of a shear instability leading to billowing structures downstream of the wave peak. We have been concerned with a detailed comparison of the experimental and numerical models of ISWs, to try and better understand the effects of breaking waves, and to quantify the mixing they produce.

People involved in this area:

• Prof. David Dritschel

Sample of our papers in this area:

Click on individual papers to go to a version of the paper available online (where open access versions are available the links should point to these).

• King, S.E., Carr, M. & Dritschel, D.G.: The steady-state form of large-amplitude internal solitary waves. J. Fluid Mech. 666, 477-505 (2011)
• Carr, M., King , S.E. & Dritschel , D.G.: Numerical simulation of shear-induced instabilities in internal solitary waves, J. Fluid Mech. 683, 263-288 (2011)

More relevant papers on the research@ St Andrews portal.