Nonlinear-resonance generated low-frequency flow

Our work at the University of Exeter (see for project details) has explored the role of fast motions (i.e. internal waves, which propagate beneath the ocean surface) in the longevity of ocean layers. These layers of different density have been frequently observed in the Arctic Ocean pycnocline, for example, during the Beaufort Gyre Exploration Project. Amazingly the layers have been observed to span almost the entire Arctic basin. The layers limit the upward flow of heat energy from the deeper ocean and essentially shield the Arctic ice from this energy. If transported to the surface waters this heat in the deep could melt the entire Arctic ice pack. Under a changing climate, changes to ocean circulation are likely to change the layer structure in the Arctic Ocean, which would change the vertical flow of heat and so affect the rate of ice melt. Therefore these layers potentially have an important role in climate change.

To explore how fast motions (internal waves) contribute to the persistence of the layers we have combined mathematical theory and numerical simulations. Although the details are complex, one key idea of the work is that the mathematics tells us that nonlinear resonance, a combination of internal waves, will cause a low-frequency signal in our system and our numerical simulations reveal this low-frequency signal to be a layer structure. Here are two plots of layers in our numerical experiments:

The plots show a layering of the fluid using contours of the vertical gradient of salinity (essentially the vertical gradient of density in this case) as a function of the vertical coordinate (y axis) and time (x axis). Near-white regions in the plots mean the fluid is well mixed. The left plot shows results from a simulation where internal waves were generated from an initial quasi-random buoyancy perturbation but with no flow and the energy in the system was then allowed to decay under diffusion. In contrast the right plot shows the results from a simulation where the flow field was continuously forced using a Gaussian shaped spectrum of wave frequencies. Both approaches have their advantages. The unforced system is simpler and we used it to help elucidate the effects of nonlinear resonance on the longevity of the layers. The signatures of internal waves are evident in the left plot, perhaps giving away their secret. It is amazing and perhaps counter-intuitive that these relatively fast moving ephemeral waves cause the long-lived slow layer structure. The right plot shows that the layers persist through time and do not decay away as they do in the left plot (note the difference in timescales between the two graphs). The Gaussian force continuously creates wave motions which via nonlinear resonance then create the low-frequency signal (the layers). This is akin to what happens in the Arctic Ocean where a spectrum of internal waves is always present, generated mainly by the tides and surface winds.

Although the project has focused on ocean phenomena, the fundamental physics and mathematics applies to other analogous systems, for instance, the atmospheres of giant planets, the hot-magnatised plasmas of fusion physics, as well as Earth's atmosphere and stratified lakes. Indeed the phenomena of nonlinear-resonance-generated low-frequency structures occurs wherever a system supports fast waves and whenever timescale separation exists. Timescale separation here refers to a system that has two processes acting simultaneously on two distinct timescales, one fast and one slow.

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