TITLE:
Pressure Gradient, Power, and Energy of Vortices
AUTHORS:
Jack Denur
KEYWORDS:
Vortex, Cyclostrophic Flow, Angular Momentum, Pressure Gradient, Geostrophic Flow, Friction-Balanced Flow, Power, Energy, Bernoulli’s Equation, Gravity
JOURNAL NAME:
Open Journal of Fluid Dynamics,
Vol.8 No.2,
June
29,
2018
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≤rmaxfluid (gas or liquid) circulates about the eye with speed v∝rn(nrmaxto 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.