Internal Gravity Waves (Eli Aghassi)

Title: Internal Gravity Waves

Principle(s) Investigated:

Density
Buoyancy
Waves
Energy transfer.

Standards (NGSS):

HS-PS4-1: Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.
HS-ESS2-1: Develop a model to illustrate how Earth’s internal and surface processes operate at different spatial and temporal scales to form continental and ocean-floor features.
HS-ESS2-5: Plan and conduct an investigation of the properties of water and its effects on Earth materials and surface processes.

Materials:

Fish tank
Salt
Food coloring
Stirring device
Scale
Divider: The divider will have to be constructed by the instructor: It should be made of a waterproof, durable, flat material (for instance PETE plastic, or even an overhead transparency) and cut to the cross-sectional size of the fish tank.
You will also need a way to keep the divider in place, for instance, tape.

Procedure:

Fill the tank with water.
Place the divider in the middle of the tank, cutting the tank into two parts. Feel free to play around with the sizes of the two parts -- they don't have to be equal.
Weigh out a large quantity of salt, and place it in one side of the tank along with food coloring. You may optionally use a different food coloring on the other side.
Stir the salt in.
Remove the divider in one smooth motion.
Optional: Sweep the divider across the surface of the water to create waves on the surface of the water. Discuss the difference between internal and surface waves
Once the wave energy has dissipated, replace the divider, this time separating approximately one third of the tank.
Mix the water in the smaller divided section. Add food coloring to the mixed section.
Remove the divider again in one smooth motion.

Student prior knowledge:

Students should understand density as it relates to fluids. This activity is ideal after students have seen and understood other activities relating to fluid stratification -- for instance of oil and water or carbon dioxide and air. This activity supplements this understanding by relating it to waves and energy transfer.

Explanation:

The waves observed are driven by the difference in density between the upper layer and the lower layer. Dissolved salt increases the density of the lower layer. In steps 7-9 we create an intermediate density layer, which propagates between the first two layers. This oscillation is driven by the buoyant force and the gravitational force.

Water waves are an oscillating system just like a pendulum or spring-block system. Any water parcel oscillates between two extremes, with energy flowing between potential and kinetic energy. We can model the energy flow of a water model, which achieves maximum gravitational potential at the crest of the wave, achieves maximum buoyant potential energy at the trough of the wave, and maximum kinetic energy at the halfway point in the middle. Unlike those other oscillating systems, however, waver waves are highly relevant to global energy flow and ocean floor erosion.

Waves between two fluids are governed by the differences in density between the two fluids. Students usually don't consider the density of the upper fluid when calculating waves (if they are asked to model waves at all). This works reasonably well for surface waves because the difference in density between air and water is so great that the air density can be approximated to zero. For internal waves between two fluids, however, this approximation completely breaks down. Instead, we can model a buoyancy-corrected (reduced) acceleration due to gravity by using the equation

where g' is the reduced acceleration due to gravity, rho is the density of the lower (heavier) fluid, rho naught is the density of the upper (lighter) fluid, and rho double naught is the characteristic buoyancy. For our purposes, the characteristic buoyancy will always be equal to rho, giving us the new equation

notice that for air (which approximates rho naught to zero) g will be equal to g'.

Angular frequency in a fluid boundary wave is determined by the Brunt-Väisälä Frequency, which is generalized to horizontally dispersing ocean waves using the equation

where z is the vertical distance between the two fluid layers. In a non-calculus context, where we are considering two discrete, non-mixing layers (which is most of the time), we should generalize this equation

where, for our purposes, z-z0 should simply be an order of magnitude approximation of the distance between two fluid layers. Ideally, for any problem, z-z0 would either be given to the student, or easily inferred from the problem setup. This is because it is not easily calculable without more detail and higher level mathematics than is appropriate for a high school class.

Recalling general equations for any type of oscillation, we can populate our model the rest of the way:

Students may be asked to model the system by using the twin resisting forces of gravity (Fg=mg') and buoyancy (Fb=-mg'). When using these forces, make sure students use the reduced acceleration due to gravity, g'. It is important to note that students should only be asked to model individual water parcels, which are analogues to discrete objects (like blocks in a spring-block system or the bob in a pendulum).

Other important discussion features include the way that internal waves shape ocean floor features by pushing and reshaping loose material on the ocean floor. Internal waves also transmit energy over enormous distances. This discussion can include the fact that, unlike surface waves, internal waves can have amplitudes of hundreds or even thousands of meters. You can make surface waves in the model to compare to the internal waves.

Questions & Answers:

When you stand on the beach, can you tell if there are internal waves present? Why or why not?

It is very difficult to see internal waves from the surface, and this usually requires very, very clear surface water and more cloudy waver underneath. Under certain conditions, however, sand kicked up by surface waves into the lower water layers create this exact scenario.

How might you surf on an internal wave?

Submarines frequently surf on internal waves. This enables them to move silently by virtually shutting down the engine. In fact, much of the research on internal waves is classified, as it is considered a security interest to know where they will be and when.

Can internal waves be dangerous?

There have been reports of scuba divers who have unknowingly crossed into the crest of an internal wave, and suddenly have been swept more than a thousand feet underwater. While a vigilant diver should be able to perform an emergency ascent to counteract the downward force, they may be swept deep underwater quickly if they are caught off-guard.

Applications to Everyday Life:

One interesting application is "dead water." When there is a boundary between two fluids at the same depth as the propeller of a boat, the boat will not move despite running the propeller. Instead, all of the energy will cause internal waves. This means that in water that looks calm and still on the surface, a boat can go to full throttle without moving anywhere.

Another application is in earthquakes. A lot of the energy that is transmitted from underwater earthquakes actually propagates as internal waves. This gives us the ability to detect seismic events, even when the surface water is calm.

A final, but very important application is nutrient mixing. Microscopic floating plants in the ocean rely on deep, nutrient rich water to survive. Without it, they die, and so do the higher level creatures in the ocean that feed on them -- including the fish we eat. Internal waves are an important and ill-understood mechanism for bringing this nutrient rich water to algae.

Photographs: Include a photograph of you or students performing the experiment/demonstration, and a close-up, easy to interpret photograph of the activity --these can be included later.

Videos:

https://www.youtube.com/watch?v=WYmRnSRsS7Y
https://www.youtube.com/watch?v=zXZZJbGteWE