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3     Background theory: Thermal Wind and Margules relation

Assuming the rotating fluid is in geostrophic balanceIn order to predict the frontal slope, one can first integrate the pressure along two different paths: one through the dense fluid and the other through the light fluid, but with the same start and end point.  Since they have the same start and end point, we require that the change in pressure calculated along these paths must be equal (see Figure 3.1).

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Figure 3.1: 

Mathinline
body\rho_{1}
 is the denser fluid, 
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body\rho_{2}
 the less dense fluid.  γ is the angle between the denser fluid and the bottom of the tank.  We integrate along paths 1 and 2, and require that the calculated pressure changes along both paths be equal.

 

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:

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