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When a vortex forms there is a linear motion of a less dense fluid rising through a descending heavier fluid. How does this linear density exchange power the rotation of a vortex?

Please note that I already understand why the rotation is favorable as it creates a tube of lower pressure at the center of rotation through which the lighter fluid rises smoothly instead of bubbling or remaining trapped below.

It seems that the linear density exchange is only capable of accelerating a pre-existing rotation, but how does it do so?

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It's the radial motion of low-density air being drawn into the upflow. Angular momentum is conserved, so as the radius decreases the angular velocity increases. For weather systems, the original angular momentum comes from the Earth's rotation.

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