Abstract
We consider small vortices, such as tornadoes, dust devils, waterspouts, small hurricanes at low latitudes, and whirlpools, for which the Coriolis force can be neglected, and hence within which the flow is cyclostrophic. Such vortices are (at least approximately) cylindrically symmetrical about a vertical axis through the center of a calm central region or eye of radius reye. In the region reye≤r≤rmax fluid (gas or liquid) circulates about the eye with speed v∝rn(n<0). We take rmax to be the outer periphery of the vortex, where the fluid speed is reduced to that of the surrounding wind field (in the cases of tornadoes, dust devils, water-spouts, and small hurricanes at low latitudes) or deemed negligible (in the case of whirlpools). If n=-1, angular momentum is conserved within the fluid itself; if n≠-1, angular momentum must be exchanged with the surroundings to ensure conservation of total angular momentum. We derive the steepness and upper limit of the pressure gradients in vortices. We then discuss the power and energy of vortices. We compare the kinetic energy of atmospheric vortices and the power required to maintain them against frictional dissipation with the same quantities for Earth’s atmosphere as a whole. We explain why the kinetic energy of atmospheric vortices must be replaced on much shorter timescales than is the case for Earth’s atmosphere as a whole. Comparisons of cyclostrophic flow with geostrophic and friction-balanced flows are then provided. We then consider an analogy that might be drawn, at least to some extent, with gravitational systems, considering mainly spherically-symmetrical and cylindrically-symmetrical ones. Generation of kinetic energy at the expense of potential energy in fluid vortices, in geostrophic and friction-balanced flows, and in gravitational systems is then discussed. We explain the variations of pressure and gravitational gradients corresponding to generation of kinetic energy equaling, exceeding, and falling short of frictional dissipation. In the Appendix, we describe a simple method for maximizing power extraction from environmental fluid (water or air) flows, which is also applicable to artificial (e.g., internal combustion) engines. In summary, we provide an overview of features and energetics of Earth’s environmental fluid flows (focusing largely on vortices) and of gravitational analogies thereto that, even though mainly semiquantitative, hopefully may be helpful.
Highlights
There are various definitions of the term “vortex”
We compare the kinetic energy of atmospheric vortices and the power required to maintain them against frictional dissipation with the same quantities for Earth’s atmosphere as a whole
We explain the variations of pressure and gravitational gradients corresponding to generation of kinetic energy equaling, exceeding, and falling short of frictional dissipation
Summary
There are various definitions of the term “vortex”. Sometimes any rotating system, or at least any rotating fluid (gas or liquid) system, is construed to be a vortex. The pressure difference between rmax and the eye is ∆Peye ≡ (P rmax ) − Peye > 0 For atmospheric vortices such as tornadoes, dust devils, waterspouts, and hurricanes, unless otherwise noted we take the fluid density ρ to be that of air at sea level or low-elevation ground level (≈1 kg/m3); for whirlpools we take ρ to be the density of water (≈103 kg/m3). We assume that horizontal (constant-altitude) changes in fluid density ρ are small enough to neglect, i.e., that, corresponding to ( ) ∆Peye ≡ P rmax − Peye , ∆ρ ρ 1 This is an excellent approximation for water in whirlpools, a very good approximation for air in dust devils and waterspouts, and a fairly good approximation for air in even the strongest hurricanes and strongest tornadoes.
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