Taylor Columns
1 Intro
One of the most interesting properties of a homogenous fluid in geostrophic flow is its ‘rigidity’-- in both the horizontal and vertical direction, the velocity vector does not vary in the direction of rotation. This is known as the Taylor-Proudman theorem, which explains the appearance of Taylor Columns, the subject of our project.
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Figure 2. A satellite photo of the Galapagos islands. Here, air is flowing from roughly left to right. The island, the “object”, is situated in the bottom left hand corner. The clouds, tracers in this case, clearly show that above the island, a column extends, breading the flow of the clouds. Behind the Taylor column, in the swirling motions, is the wake caused by the islands Taylor column.
2 Theory (The Taylor-Proudman Theorem)
The Taylor-Proudman Theorem says that for a fluid meeting the conditions stated above, velocity will not vary in vertical direction. To derive this conclusion, we will start with the momentum equation of a fluid in geostrophic balance with negligible friction where the Coriolis force, pressure gradient, and gravity are balanced, which is
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By hydrostatic balance, ∂p/∂z can be rewritten as -ρg, and if ρ and f are constant, then their derivatives with respect to position are zero. Therefore, the horizontal wind does not vary with height.
3 Experiment
There were two parts to our experiment: the first was to test whether we could replicate Taylor columns with an object placed in a rotating tank and the second was to test whether there would still be a disruption when an object was placed in a tank with water that had a temperature gradient
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Figure 5. A video of experiment 2, showing that the density gradient causes the Taylor column to tilt.
4 Thermal Wind
To begin to understand the relationship between Taylor columns and thermal wind, it is important to understand isothermal surfaces, which is a surface where the temperature of the fluid or air is equal everywhere. Isothermal surfaces can be made in both the lab and exist in the atmosphere. These surfaces are, in both rotating fluid labs and in the atmosphere, three dimensional surfaces. Lets first look at an example of a surface visible in a lab.
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