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3 Background theory: Thermal Wind and Margules relation
Assuming the rotating fluid is in geostrophic balance, one can integrate the change in pressure around a closed loop at the interface of the two different density fluids and require it to be 0, resulting in the following relation (Margules relation) for the slope of the frontal boundary:
where γ is the angle the dense fluid makes with the horizontal,
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Assuming that angular momentum of the fluid is conserved (only a valid assumption when friction can be discounted), we get the following relation for the (approximate) deformation radius:
, where H is the height of the free surface with respect to the bottom of the tank.
The deformation radius gives an idea of how much the front will spread before reaching a stable state.
4 Lab results
In order to collect data from our experiment, we tracked using video processing software buoyant particles at the surface of the fluid, as well as particles with density in between that of the two fluids which sat just above the frontal boundary. The shape of the front, although relatively stable, did vary slightly throughout the course of the experiment, as can be seen from the following images.
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Using the equator for the deformation radius,
we calculate the predicted radius of deformation to be around 20 centimeters, or half of the width of the tank. Since the boundary of the front extends all the way to the outside of the tank, about 3/4 of the distance to the bottom of the tank, the actual radius of deformation for this front is slightly larger than this prediction. Therefore, we see the slope of the front decrease towards the edge of the tank, as it has no more room to spread outwards.
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