Abstract
For obvious reasons, geophysicists, oceanographers and planetary physicists use a coordinate system that rotates with the planet: the co-rotating frame . To describe the medium (the ocean, the atmosphere or the magma in the Earth’s interior), they have to use the equations outlined in the preceding section. In many applications, the vorticity of the fluid or gas plays an important role. Since we have transformed the velocities to the co-rotating frame, something analogous must be done for the definition of the vorticity, and the associated equation of motion. This will lead to the definition of the absolute vorticity , whose definition includes a contribution from the swirling motions in the rotating frame as well as a contribution from the planetary rotation.
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- 1.
An interesting exercise in vector analysis, best performed in cylindrical coordinates with the rotation axis chosen along the z-axis.
- 2.
The rotation periods are for Jupiter: 0.41 day, for Saturn: 0.44 day and for Uranus: 0.65 day.
- 3.
see for instance Ref. [37], Chap. 3.
- 4.
There is an analogous velocity in classical mechanics: a pendulum with length \(\ell \), suspended in a gravity field with a uniform gravitational acceleration g, oscillates with frequency \(\omega = \sqrt{g/ \ell }\) around the vertical for small-amplitude oscillations. The velocity of the mass at the end of the pendulum equals \(v=\omega \ell = \sqrt{g \ell }\).
- 5.
Because of mass conservation, the material must flow away again at high altitude, leading to an anti-cyclonic circulation at great height.
- 6.
At great depth, Jupiter probably has a solid core of rocks, surrounded by metallic hydrogen.
- 7.
In contrast: the cyclones and hurricanes in the Earth’s atmosphere typically last for a period of the order of weeks!
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Achterberg, A. (2016). Fluids in a Rotating Frame: Applications. In: Gas Dynamics. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-195-6_13
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DOI: https://doi.org/10.2991/978-94-6239-195-6_13
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