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Vortices on a Rotating Sphere with Layers of Fluid on its Surface |
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The ultimate purpose of this analysis is to explain the wind velocities in tropiical cyclones (hurricanes, typhoons, etc.) and tornadoes. It involves the draining of fluid from one level to another with the preservation of angular momentum.
Consider a horizontal disk of radius R and thickness D rotating at an angular rate of ω. Now consider a horizontal arrow placed through the center of disk and a parallel arrow through any other point of the disk. As the disk rotates the arrows remain parallel. Thus the rates of rotation of the arrows are the same for the one at the center and through any other point in the disk.
Now consider a small circular area within the disk. Let the distance of the center of this circular area from the center of the disk be r and the radius of circular area be ρ. The circular area has a rate of angular roration about its equal to the rate of roration of the disk about its center; i.e., &omega. Thus the circular area has a mass M, a moment of inertia J and an angulr momentum L given by
where γ is the mass density of the material of the disk.
Now suppose the circular area contracts from a radius ρ to a radius ρ' while maintaining the mass M and angular momentum L. Then
This is aa model of what happens with tropical cyclones like hurricanes and typhoons.
Almost everyone has seen iceskaters go into a spin with arms outstretched and then pull their arms close to their bodies and go into noticably faster spins. This illustrates the preservation of angular momentum. Angular momentum L is equal to the product of moment of inertia J times the angular rate of rotation ω, i.e.,
If L is to remain constant then when J goes down ω has to go up.
A mass of m rotating about a center at a distance ρ has a moment of inertia of J=mρ²
Therefore
But tangenial velocity v is equal to ρω thus
Consider a drain pan with a diameter of 20 inches and a drain pipe with a diameter of 1 inch. The angular rate of rotation of the flow through the drain pipe would be 400 times the rate in the pan. The tangential velocity of the fluid in the pipe would be 20 times that in the pan.
Consider a hurricane with an overall width of 300 miles and an eye with of 30 miles. Thus the tangential wind velocity at the wind wall at the eye should be ten times the wind velocity at the periphery of the hurricane. If the wind velocity at the periphery is 15 miles per hour at the wind wall it would be 150 miles per hour.
Let a sphere of radius R be rotating at an angular rate of ω. Any circular region centered on the North Pole would also be rotating at an angular rate of ω. But a circular region centered on the equator of the sphere would not rotate about its center at all. However in that case there would be rotation about the north-south axis, The angular rate of rotation of the region about that axis would also be ω.
Consider a circular region at a midlatitude of θ in the northern hemisphere. Let the radius of the region be denoted as rho;. The area and position of the region can be represented as a vector perpendicular to its surface through its center of magnitude equal to iits area of πρ².
That area vector can be decomposed into two perpendicular vectors; one parallel to the spin axis of the sphere corresponding to a circular region of radius equal to ρ_{0}=ρcos(θ). The other vector is perpendicular to the spin axis of the sphere and of magnitude corresponding to a circle of radius ρ_{0}=ρsin(θ).
As the sphere turns both circular regions undergo the same angular rotation as the sphere. Thus their rotation rates are both &omega .However it is only the second circular region that is relevant for the dynamics of tropical cyclones.
A cyclone arises from region having higher temperature and humidity than the atmosphere above it. The warm, moist air finds a path of least resistance and forms a funnel into the cool, dry air above lt. As the warm, moist air drains upward it draws in air from a distance. Because angular momentum is preserved the region increases its rate of rotation. From the previous analysis any region on Earth except at the equator has some inherent rotation to be amplified due to the flow of air toward its center. Thus a cyclone is created.
(To be continued.)
On a rotating sphere any region except at the equator has an inherent angular momentum that can be amplified by flows between layers on its surface.
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