Math and water
The application of mathematical models ranges greatly from forecasting finances to skull surgery to climate adaptation. Applied mathematics professor Kevin Lamb is using mathematical models to determine how energy is transferred between internal and non-linear waves in marine systems.
Leveraging the use of theoretical and computational methods, Lamb primarily looks at internal waves occurring beneath the surface of lakes and oceans.
“An example of an internal wave would be when you go swimming and it’s really warm at the surface and you dive down and it gets much colder, and you hit the thermocline – well you can have waves on the thermocline. In the ocean, internal waves exist because of salinity variations and temperature,” explained Lamb. “They’re driver by winds and by the interaction of tides with topography.”
A thermocline, or metalimnion in lakes, is a thin layer found in a large body of water in which temperature changes rapidly with depth. This layer moves vertically in systems depending on season, topography, wind, and salinity.
“Waves on the thermocline will shoal and break, just like surface waves, and are important to ecology. Internal waves promote nutrient mixing and heat distribution across the thermocline,” said Lamb. “
Most of Lamb’s work is done with numerical models and simulations that help explain physical processes. Lamb admits, working with internal waves can be trying at times.
"Non-linear waves – they’re very complicated. All the different waves are coupled together and energy is transferred from one wave to another. In the real world, the topography can be rather complicated,” said Lamb.
Theorizing about the ideal system and situation is one thing but the application comes when you have the equations to back your theories up. “You get a lot of important insight from theoretical work but you really need to do the numerical work so you can understand what these waves are doing.”
Lamb’s models have application in a variety of situation and studies. Understanding how energy is transferred through internal wave fields, how waves behave as they interact with topography, their physical processes and their implications, is essential.
“In the ocean, some of these waves are very large and have large currents which can impact structures. The oil industry is quite interested in understanding these waves,” said Lamb. “Breaking waves resuspend sediments off the bottom [of the ocean] and can erode the continental shelf slope. Resuspending sediments also serves as an important source of nutrients and because of that, they can also impact oxygen demand.”
Understanding how the energy is transferred from large to small scale, over time, is the primary focus of Lamb’s research. From there, others can leverage information and associated models.
“People doing climate modeling want to know where the mixing occurs so they can get the right overall circulation patterns and that mixing is driven by internal waves so you need to know where does the energy go, where does the mixing occur?”
Currently, Lamb is looking at non-linear waves in the ocean that can reach 200 metres tall in the South China Sea. The research team is looking at the implications of how these massive internal waves break and shoal as they reach shallow water.
For his research, animations are essential. “I like to generate animations. It really helps you understand what’s going on. I also make a lot of them for the courses I’m teaching. It’s crucial. Animations give you a lot of insight into what is going on in these big numerical simulations.”