Pressure Gradient, Power, and Energy of Vortices

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 eye r . In the region eye max r r r ≤ ≤ fluid (gas or liquid) circulates about the eye with speed n v r ∝ ( 0 n < ). We take max r 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, waterspouts, and small hurricanes at low latitudes) or deemed negligible (in the case of whirlpools). If 1 n = − , angular momentum is conserved within the fluid itself; if 1 n ≠ − , 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 How to cite this paper: Denur, J. (2018) Pressure Gradient, Power, and Energy of Vortices. Open Journal of Fluid Dynamics, 8, 216-249. https://doi.org/10.4236/ojfd.2018.82015 Received: April 27, 2017 Accepted: June 26, 2018 Published: June 29, 2018 Copyright © 2018 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

r . In the region eye max r r r ≤ ≤ fluid (gas or liquid) circulates about the eye with speed n v r ∝ ( 0 n < ). We take max r 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, waterspouts, and small hurricanes at low latitudes) or deemed negligible (in the case of whirlpools). If 1 n = − , angular momentum is conserved within the fluid itself; if 1 n ≠ − , 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

Introduction
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. For our purposes let us construe a vortex to be any rotating fluid (gas or liquid) system wherein the speed v of fluid motion increases monotonically inwards from the outer periphery max r towards decreasing radial distance r from the axis of rotation, i.e., v increasing monotonically with decreasing r, attaining a maximum value max v at the circumference of a calm area or eye of radius eye r about the axis of rotation. (This monotonic increase of v with decreasing r in numerous instances of real vortices is interrupted by local fluctuations, but in such instances it is the secular trend that we focus on.) Thus we construe tornadoes, dust devils, waterspouts, hurricanes, and whirlpools to be vortices, but not rotating fluid systems that lack an eye such as at least the vast majority of extratropical cyclones if not all of them, and all anticyclones. In short, we construe a vortex to be a cyclone with an eye. Our main interest concerning fluid (gas or liquid) systems will be in those meeting our construed definition of "vortex", but we will also consider in some measure fluid systems not meeting this definition.
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 [1] [2]. Because waterspouts are intermediate in size, intensity, and lifetime between dust devils and tornadoes, their properties can be interpolated between those of dust devils and tornadoes, so we provide only limited consideration of them.
Even small hurricanes are much larger than even the largest tornadoes, let alone than waterspouts and dust devils, but they are still small enough that, especially at low latitudes, the Coriolis force can be neglected, at least in their maintenance. We do not consider their initial formative stages, in which the Coriolis force, even though small at low latitudes, is nevertheless important. The centripetal (sometimes construed as centrifugal) force is much smaller than the Coriolis force in the initial formative stages of hurricanes, even of small ones at low latitudes. But in fully-formed hurricanes, especially small ones at low latitudes, the reverse is true (referring to the first four paragraphs of Section 2 may be helpful). We employ the term "hurricanes" to encompass all tropical cyclones In such vortices, the balance of forces on any parcel of moving fluid (gas or liquid: in the cases considered, air or water, respectively) can be considered cyclostrophic [1] [2] with negligible error [1] [2]. r about an axis through its center be increased. In this case, v increases while r remains fixed; thus n = −∞ .] Note that (not within our construed definition of "vortex") 0 n = corresponds to constant v (v independent of r), and that 1 n = + corresponds to solid-body (wheel-like) rotation. As we construe vortices to be cyclones with eyes, minimum pressure obtains in the eye, with pressure increasing monotonically with increasing r, i.e., 0 P r ∂ ∂ > , in the region eye max r r r ≤ ≤ . Let the sea-level or ground-level pressure at the outer periphery of a vortex be ( ) max P r and that in the eye be eye P .
Of course ( ) eye max P P r < . The pressure difference between max r and the eye is 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/m 3 ); for whirlpools we take ρ to be the density of water (≈10 3 kg/m 3 ). We assume that horizontal (constant-altitude) changes in fluid density ρ are small enough to neglect, i.e., that, corresponding to  . 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. Indeed for whirlpools also neglecting vertical variations in water density ρ is an excellent approximation (All pressures and densities are at sea level or low-elevation ground level unless otherwise noted.) If, as in the cases of most interest to us as per our construed definition of "vortex", In Section 2, we discuss cyclostrophic flow, and derive the steepness and upper limit of the pressure gradient in vortices. In Section 3, we 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. The effects on Earth's atmosphere as a whole of a cutoff of insolation, and of its partial cutoff in the winter hemisphere, are discussed. Comparisons of cyclostrophic flow with geostrophic and friction-balanced flows are provided in Section 4. In Section 5 we consider an analogy that might be drawn, at least to some extent, with gravitational systems. We consider mainly spherically-symmetrical and cylindrically-symmetrical gravitational systems. Generation of kinetic energy at the expense of potential energy in cyclostrophic flow of fluid vortices, in geostrophic and friction-balanced flows, and in gravitational systems is discussed in Section 6. We explain the variations of pressure and gravitational gradients corresponding to generation of kinetic energy equaling, exceeding, and falling short of frictional dissipation. Concluding remarks are provided in Section 7. In the Appendix, we describe a simple method for maximizing power extraction from environmental fluid (water or air) flows. We also briefly explain the application of this method 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.

Cyclostrophic Flow, and Steepness and Upper Limit of the Pressure Gradient in Vortices
If our small fluid parcel is moving on a circular path about the center of an eye, at radial distance r from the center of the eye, at speed v, then the centripetal force required to keep it on this circular path is 2 centr mv F r = .
(2) (Sometimes centr F is alternatively construed as the centrifugal force that the fluid parcel experiences and that balances the pressure-gradient force grad Strictly, grad F is attractive towards the center of a vortex and therefore negative, but we are concerned mainly with its magnitude, so we omit the minus sign.

Now let
where ( ) max eye r r r ρ is the average of ( ) r ρ in the range eye max r r r ≤ ≤ at the altitudes where v and G are measured, most typically 10 m above the surface and at sea level or ground level, respectively. (Enclosure within angular brackets denotes the average of the enclosed quantity.) Since ρ decreases radially inwards, the functional dependency of ( ) r ρ on r, considered alone, results in G decreasing with decreasing r. The approximation in the last term of Equation (5), which neglects this functional dependency, is justified because, in accordance with the third-to-last paragraph of Section 1, corresponding to ( ) eye max eye P P r P ∆ ≡ − 1 n = − is typically at least a close approximation-tornadoes, dust devils, waterspouts, and whirlpools-G increases very rapidly inwards, i.e., 3 G r − ∝ , in the region eye max r r r ≤ ≤ [5]. (Reference [5] cites an excellent late 19 th -centry work FINTIA for in-depth explanations.) In typical hur- , with the most typical value perhaps 2 3 n ≈ − . Corresponding to 3 4 n = − , and corresponding to Let us briefly consider the range 0 n ≥ that is not within our construed definition of "vortex", assuming the approximation in the last term of Equation (5).
Note that 0 n = , i.e., v independent of r, corresponds to In the last term of Equation (7) (8) and (9) are better than those in Equations (6) and (7).] Again, Equations (6)-(9) represent theoretical upper limits, neglecting frictional losses, and hence corresponding to angular momentum being conserved within the fluid itself; i.e., to 1 n = − . Owing to frictional losses, in actual vortices n is typically at least very slightly larger than −1, and hence attainable values of v and G are typically at least very slightly smaller than those given in Equations (6)- (9). (In rare cases vortices may generate angular momentum and kinetic energy fast enough to more than offset frictional losses; hence in these rare cases

Very Small Atmospheric Vortices: Tornadoes, Dust Devils, and Waterspouts
The solar constant at Earth is ≈1400 W/m 2 . Over day and night, over all four Open Journal of Fluid Dynamics seasons, and over clear and cloudy weather, the average solar power flux density absorbed (and thence reradiated) by Earth's surface is ≈200 W/m 2 . Of this ≈200 W/m 2 , ≈1% or ≈2 W/m 2 is converted into wind power flux density. The power required to maintain wind speed v against friction is, at least approximately, 3 v ∝ . A fair estimate of the root-mean-cube average surface wind speed on Earth (at the official anemometer elevation of 10 m above Earth's surface assuming no obstructions) is ≈5 m/s. Thus a fair estimate of the power flux density ℙ required to maintain surface wind speed of v m/s is [There is some questioning of the Betz limit concerning vertical-axis windmills [9] [10]. A simple method for maximization of power extraction from environmental fluid (water or air) flows is discussed in the Appendix.] Assuming extraction of , ≈14 m 2 of Earth's surface are required to supply each 1 m 2 of windmill at the official anemometer elevation of 10 m above Earth's surface assuming no obstructions. Considering a windmill at a higher elevation, say ≈200 m above Earth's surface, where, say, rmc 8 m s v ≈ , Earth's surface than to ≈1 km above it.) Since we seek only approximate results we can take rmc,surface 5 m s v ≈ to be representative. We take the outer radius max r of a tornado or dust devil to be that at which the surface wind (at the official anemometer elevation of 10 m above Earth's surface assuming no obstruc-  (10) and (11) the total power vortex vortex vortex S =  P required to maintain a tornado's or dust devil's wind against friction, and hence also the frictional dissipation, is is the root-mean-cube average wind speed at 10 m within eye max r r r ≤ ≤ . (Note: Don't confuse these three symbols: P = pressure,  = power flux density, P = power.) Hence the total energy required to maintain a tornado or dust devil for the duration of its lifetime lifetime τ , and ultimately frictionally dissipated, is Since a tornado or dust devil is typically at least largely confined to the lower ≈1 km of Earth's atmosphere, whose mass is ≈ 1000kg per m 2 of Earth's surface, its kinetic energy is where is the root-mean-square average wind speed at eye max r r r ≤ ≤ within the tornado or dust devil. Thus the kinetic energy of a tornado's or dust devil's winds must be replaced on a timescale In the third steps of Equations (15) and (16) we applied the approximation

Larger (But Still Small) Atmospheric Vortices: Small Hurricanes at Low Latitudes
Now consider small hurricanes at low latitudes. Even the smallest hurricanes are much larger than even the largest tornadoes, let alone than even the largest waterspouts or largest dust devils, but they are still small enough that, especially at low latitudes, the Coriolis force can be neglected, at least in their maintenance.
We do not consider their initial formative stages, in which the Coriolis force, even though small at low latitudes, is important. The centripetal (sometimes construed as centrifugal) force is much smaller than the Coriolis force in the initial formative stages of hurricanes, even of small ones at low latitudes. But in fully-formed hurricanes, especially small ones at low latitudes, the reverse is true.
(Referring to the first four paragraphs of Section 2 may be helpful.) So we limit our considerations to fully-formed small hurricanes at low latitudes, for which the Coriolis force can be neglected, and hence for which the balance of forces on any parcel of moving air can be considered cyclostrophic [1] [2] with negligible error [1] [2]. We employ the term "hurricanes" to encompass all tropical cyclones of this type, e.g., including Pacific typhoons, although we will occasionally refer to Pacific typhoons specifically.
The majority of the kinetic energy of hurricane circulations is typically within the lower half of Earth's troposphere. A fair estimate of the root-mean-cube average wind speed within the lower half of the low-latitude (tropical) troposphere is rmc,trop 10 m s v ≈ . Recalling the first two paragraphs of Section 3.1, the power flux density required to maintain wind speed v within the lower half of the low-latitude troposphere is The power flux density of ≈2 W/m 2 maintaining Earth's winds against friction We take the outer radius max r of a small low-latitude hurricane to be that at which the surface wind speed (at the official anemometer elevation of 10 m above Earth's surface in the absence of obstructions) is surface 5 m s v = . Thus we is the root-mean-cube average wind speed at 10 m within eye max r r r ≤ ≤ . Hence the total energy required to maintain a small low-latitude hurricane for the duration of its lifetime lifetime τ , and ultimately fric- Since the majority of the kinetic energy of the circulation of a small low-latitude hurricane is typically within the lower half of the troposphere, whose mass is ≈5 × 10 3 kg per m 2 of Earth's surface, its kinetic energy is In the third steps of Equations (22) and (23) we applied the approximation ≈ × , as for exceptionally intense and/or long-lived hurricanes, and even more so for exceptionally intense and/or long-lived western Pacific typhoons. [Of course, even at moderately low latitudes, the Coriolis force may not be very small compared to the centripetal (sometimes construed as centrifugal) force in the outer regions of exceptionally large hurricanes and even more so of exceptionally large western Pacific typhoons.] We note that even the most active earthquake faults are doing well to manage one magnitude-7 (or larger) earthquake every few decades [11], and one magnitude-7 (or larger) earthquake per century or longer is more typical [11].

Comparison of Earth's Atmospheric Vortices with Earth's Atmosphere as a Whole
For Earth's atmosphere as a whole, the root-mean-cube average wind speed is Neglecting the difference between the root-mean-cube and root-mean-square average wind speeds, the total kinetic energy of wind per m 2 of Earth's surface is Thus the replacement timescale for the kinetic energy of Earth's winds is FTNT7A The quantity Of course, a partial cutoff of insolation befalls the winter hemisphere of Earth.
But as per Equation (26), atmospheric thermodynamic efficiency is on the whole higher in winter than in summer, because temperature differences between oceans and continents at any given latitude, and between low latitudes and high latitudes (substitute subscripts: ocean → lowlat and cont → highlat), are greater in winter than in summer. Increased atmospheric thermodynamic efficiency more than compensates for decreased insolation (decreased E), so (excepting convective weather systems, e.g., thunderstorms and hurricanes) on the whole  is greater and hence atmospheric circulation is more vigorous in winter than in summer.

Whirlpools
A whirlpool in a sink is powered at the expense of the gravitational potential energy of the water. The maximum water speed, at the bottom of the eye wall at the drain, is (if frictional losses are negligible) ( ) where 3 3 10 kg m ρ ≈ is the density of water, g is the acceleration due to gravity, H is the height of the water surface (above the floor of the sink) far from the whirlpool [ Hence if the water is not replaced the e-folding time of a whirlpool is

Geostrophic and Friction-Balanced Flows
Although our main concern in this paper is with cyclostrophic flow, comparisons with geostrophic flow (straight isobars) FTNTS9,10 , and with friction-balanced flows, may be edifying.
For geostrophic flow, Equation (1) where max H ∆ is the dip of the isobaric surface that is at sea level or ground level at the locations of minimum surface barometric pressure min P from its altitude at the locations of maximum surface barometric pressure max P [ respectively. In Equations (34) and (35)  is even at worst a fairly good approximation for all geostrophic (or quasi-geostrophic) flows in Earth's atmosphere, and always an excellent approximation for all geostrophic (or qua- Geostrophic (or quasi-geostrophic) flow is quite common from latitude 20 φ ≈  polewards, both in Earth's atmosphere and in its oceans. Of course, in Earth's lower atmosphere, 3 1 kg m ρ ≈ , whereas in its oceans, Geostrophic and balanced FTNT1B anticyclonic flows become increasingly difficult to maintain as φ decreases below ≈20˚, and are impossible to maintain at the equator. Thus in approaching the equator from latitude 20 φ ≈  , these flows become progressively more friction-balanced, until at the equator they must be purely friction-balanced. Friction-balanced flow can also occur even at higher latitudes in, for example, the damming of cold waves by mountain ranges. FTNT11

si-geostrophic) flows in Earth's oceans. Thus the approximations in Equations
Another example of friction-balanced flow is river flow FTNT12 . The force driving the flow of a river, per unit mass of flowing water, is driv sin interaction with its river bed, this retarding frictional force, per unit mass of flowing water, is also, at least approximately, proportional to C′  , where C′ is a dimensionless factor accounting for, say, the roughness of the river bed,  is the river bed's wetted perimeter, and  is the river's cross-sectional area. Thus Most typically, sinθ decreases downstream, but C′   increases slightly faster downstream, so v increases slightly downstream. (As previously noted, most often for river flows driv 1 rad sin F m g Yet another example of friction-balanced flow is the flow of groundwater. The force driving the flow of groundwater, per unit mass, is driv sin g is the acceleration due to gravity and θ is the hydraulic gradient, i.e., the slope Note that Darcy's Law FTNT13 in hydrology [12] [13] is essentially equivalent to Ohm's law FTNT14 in electrical circuits [13]. Groundwater flows occur at much smaller Reynolds numbers than river-water flows, atmospheric winds, and most oceanic flows. At small Reynolds numbers (as for groundwater flows, with rare exceptions FTNT13 ) viscous drag is predominant so, at least approximately, fric F v ∝ : viscous shear forces are, at least to a first approximation, proportional to v [14] [15]. At large Reynolds numbers (as for atmospheric winds, river-water flows, and most oceanic flows), inertial drag is predominant, so, at least to a first approximation,

Is There an Analogy?
There is, at least superficially, similarity between the spiral rainbands of hurricanes and the spiral arms of a galaxy such as our own Milky Way. Air orbits about the eye in a hurricane. Stars, including the Sun, as well as gas, dust, etc., orbit about the center of the Milky Way, and of course orbital motion can occur about any gravitating body. Thus can a galaxy such as our own Milky Way, or any gravitating system in general, be in any way construed as cyclonic? (Clearly a galaxy such as our own Milky Way, or any gravitating system in general, cannot be construed as anticyclonic, because gravity is an attractive force, and the pressure-gradient force grad F is attractive in cyclones but repulsive in anticyclones.) Can an analogy be drawn? A cute little book [16], even though scientifically inaccurate, at any rate suggested this at least superficial similarity, and hence the possibility of an analogy, to the author of this present paper. Such an analogy may also seem suggestive based on more recent, scientifically accurate, books [17] [18] [19]. The similarity between not only the spiral rainbands of hurricanes and the spiral arms of a galaxy such as our own Milky Way, but also between Open Journal of Fluid Dynamics both and other structures (albeit not with respect to either the Milky Way or these other structures being construed as cyclonic) has been noted by authors of other more recent, scientifically accurate, books [20] as well FTNT15 .
Perhaps an analogy can be drawn, at least to some extent. Recall that in a cyclone, minimum pressure occurs at the center (in the eye if the cyclone has one and hence is a vortex as per our construed definition in the first paragraph of Section 1), and pressure increases monotonically with increasing r (in the region eye max r r r ≤ ≤ if the cyclone has an eye, and perhaps somewhat beyond max r ). But the pressure in the intergalactic space surrounding the Milky Way, and indeed in the space surrounding any gravitating body, is for all practical purposes zero. Thus if the Milky Way, and indeed any gravitating body, is to be construed as cyclonic, then the pressure within its domain must be construed as negative, decreasing radially inwards, and most strongly negative at its center. Hence gravity must be construed as generating a negative pressure or tension.
If gravity generates tension, then space must be capable of supporting tension. If space is construed as a medium rather than as mere nothingness, then perhaps this tension could be construed as warping or curving space. Perhaps this might provide a physical interpretation for the statement: "Spacetime tells matter how to move, matter tells spacetime how to curve [21]" . [Whereas the entirety of spacetime tells matter how to move and matter tells the entirety of spacetime how to curve [21], pressures (including tensions) and pressure gradients (including tension gradients) are purely spatial FTNT15A , although of course they can evolve with time FTNT15A . In this Section 5, we consider only unchanging gravitators, and hence only unchanging pressures (specifically tensions) and only unchanging pressure gradients (specifically tension gradients) FTNT15A .] For, how can nothingness tell matter how to move, and how can matter tell nothingness how to curve? Does the phrase "curvature of nothingness" even have any meaning? Perhaps the classical vacuum might be construed as nothingness, but the quantum-mechanical vacuum certainly cannot [22] [23] [24]. Moreover, since a medium capable of supporting tension is required for the transmission of transverse waves [by contrast, longitudinal waves, e.g., sound, can travel through any (material, i.e., nonvacuum) medium], and since electromagnetic radiation is comprised of transverse waves, can space be construed as a latter-20th-century and 21st-century interpretation of the ether [sometimes spelled aether (the a is silent)] postulated in 19th-century physics [25] [26] FTNT15B ? Concerning the latter point, the conventional viewpoint is, of course, that electromagnetic waves serve as their own medium-their own ether-via the continual handoff of energy from transverse electric field to transverse magnetic field to transverse electric field… [27].

Spherically-Symmetrical Gravity
For an isolated nonrotating spherically-symmetrical gravitator of radius r * and mass M for which Newtonian theory is sufficiently accurate for calculation of Strictly, grav F is attractive towards the center of a gravitating body and therefore negative, but we are concerned mainly with its magnitude, so we omit the minus sign.
Radial spatial intervals  of unity ( For weak spherically-symmetrical gravitational fields at * r r ≥ Qualitatively, we should expect that if tension, i.e., negative pressure, is effected by an isolated nonrotating spherically-symmetrical gravitator, then such tension would radially stretch space, but leave unaltered space perpendicular to the radial, i.e., leave unaltered the Euclidean ruler-distance measure 2πr of any circumference and the Euclidean (ruler-distance)² measure 4πr 2 of any spherical shell about the center of the gravitator FTNTS15A,17-17B . [Also of course time is dilated radially inwards FTNTS15A,17-17B , in the weak-field limit as per the plus (+) signs in Equations (41) and (42) being replaced by minus (−) signs FTNTS15A,17-17B , but we focus on the spatial, specifically spatial radial, gravitational modifications of spacetime FTNTS15A,17-17B .] Qualitatively, this radial stretching of space seems con-Open Journal of Fluid Dynamics sistent with any circumference and any spherical shell about the center of an isolated nonrotating spherically-symmetrical gravitator whose respective Euclidean ruler-distance and (ruler-distance)² measures are 2πr and 4πr 2 possessing a radius whose ruler-distance measure exceeds the Euclidean value r [in the weak-field limit by approximately the ratio given by Equations (41) and (42)] FTNTS15A,17-17B . Quantitatively, we may be on less certain ground if we try to relate ( ) P r ∆ to the degree of radial stretching of space FTNTS15A,17-17B , but let us try anyway.
Can we draw the following analogy at r r * ≥ , as per Equations (6) and (8), with the help of Equations (39) and (42), letting r ρ ≤ be the average density within r?  (4) and (5): Thus perhaps our analogy can be drawn, at least to some extent. But we cannot expect more than qualitative validity from our simplified, or even oversimplified, analyses. Yet it should be noted that an elastic-strain theory of gravity has been considered on a much more rigorous level [29] FTNT22 .

Cylindrically-Symmetrical Gravity
For comparison, let us consider (in the Newtonian approximation) the gravitational field of an isolated nonrotating long cylindrical mass M of radius r * and finite length * l r  , at radial distance r from its central axis and at the center of its length ( 2 l from both ends), with * 2 r l r ≤  .
In Equation ( Thus not even the most powerful rocket, indeed not even light, can escape from an infinitely long sewing thread! Thus if l → ∞ there is no Newtonian (weak-field) limit for a cylindrically-symmetrical gravitational field, not even for that of a sewing thread. For finite l (as stipulated in the first sentence of this Section 5.3) in the weak-field limit grav,cyl F gradually changes from that given by Equation (45)  Thus, again, perhaps our analogy can be drawn, at least to some extent. But, again, we cannot expect more than qualitative validity from our simplified, or even oversimplified, analyses. Yet, we again note that an elastic-strain theory of gravity has been considered on a much more rigorous level [29] FTNT22 .

Generation of Kinetic Energy in the Cyclostrophic, Geostrophic, Friction-Balance, and Gravitational Cases
In order to generate kinetic energy in cyclostrophic fluid flow, the fluid must be able to spiral inwards down a hill, or rather down into a pit, of pressure, crossing isobars towards lower pressure, so that the potential energy represented by high pressure can be traded for kinetic energy at lower pressure, in accordance with Bernoulli's equation of energy conservation for fluid flow [6] [7] FTNT2 , as per Equations (6) and (8). But in order to spiral inwards down a hill, or rather down into a pit, of pressure, there must be friction. In the absence of friction the fluid would simply orbit at fixed r always instantaneously parallel to the isobars and consequently with fixed v, and hence would never be able to spiral inwards down a hill, or rather down into a pit, of pressure. If in cyclostrophic flow G increases with decreasing r as 1 r − , generation of kinetic energy matches frictional loss so v is constant, independent of r. If in cyclostrophic flow G increases with decreasing r faster than as 1 r − , generation of kinetic energy exceeds frictional loss so v increases with decreasing r. If in cyclostrophic flow G increases with decreasing r more slowly than as 1 r − , generation of kinetic energy falls short of matching frictional loss so v decreases with decreasing r. [This presumes the approximation in the last term of Equation (5). Refer to Sections 1 and 2 as necessary.] Although our main concern in this paper is with cyclostrophic flow, comparisons with generation of kinetic energy in geostrophic flow (straight isobars) FTNTS9,10 , in friction-balanced flows, and also in gravitational cases, may be edifying (Refer to Sections 4 and 5 as necessary.) In order to generate kinetic energy in geostrophic fluid flow, the fluid must be Open Journal of Fluid Dynamics able to move down a hill of pressure, crossing isobars towards lower pressure, so that the potential energy represented by high pressure can be traded for kinetic energy at lower pressure, in accordance with Bernoulli's equation of energy conservation for fluid flow [6] [7] FTNT2 , as per Equations (32) and (34). But in order to move down a hill of pressure, there must be friction. In the absence of friction the fluid would simply move at fixed pressure always parallel to the isobars and consequently with fixed v, and hence would never be able to move down a hill of pressure. If in geostrophic flow sin G φ is constant (independent of pressure), generation of kinetic energy matches frictional loss so v is also constant (independent of pressure). If in geostrophic flow sin G φ increases in the direction of decreasing pressure, generation of kinetic energy exceeds frictional loss so v also increases in the direction of decreasing pressure. If in geostrophic flow sin G φ decreases in the direction of decreasing pressure, generation of kinetic energy falls short of matching frictional loss so v also decreases in the direction of decreasing pressure.
In friction-balanced fluid flows, the fluid is always able to move down a hill of pressure or of elevation, crossing isobars towards lower pressure or contours towards lower elevation, so that the potential energy represented by high pressure or high elevation can always be traded for kinetic energy at lower pressure or lower elevation. If in friction-balanced atmospheric or oceanic flow G C is constant (independent of pressure), generation of kinetic energy matches frictional loss so v is also constant (independent of pressure). If in friction-balanced atmospheric or oceanic flow G C increases in the direction of decreasing pressure, generation of kinetic energy exceeds frictional loss so v also increases in the direction of decreasing pressure. If in friction-balanced atmospheric or oceanic flow G C decreases in the direction of decreasing pressure, generation of kinetic energy falls short of matching frictional loss so v also decreases in the direction of decreasing pressure.
All river and groundwater flows are friction-balanced flows: thus in these flows the water is always able to move downhill, so that the potential energy represented by high elevation can always be traded for kinetic energy at lower elevation. If in river flow ( ) sin C g θ ′ ÷   , and hence also v, increases slightly downstream.] If in groundwater flow sin C g θ ′′ is constant, generation of kinetic energy matches frictional loss so v is also constant (independent of elevation). If in groundwater flow sin C g θ ′′ increases downhill, generation of kinetic energy exceeds frictional loss so v also increases downhill. If in groundwater flow sin C g θ ′′ decreases downhill, generation of kinetic energy falls short of matching frictional Open Journal of Fluid Dynamics of kinetic energy exceeds frictional loss so v increases as a satellite spirals inwards towards decreasing r. By contrast, in the vicinity of the Sun's orbit in the Milky Way, or in the case of cylindrically-symmetrical gravitation, 1 G r − ∝ , hence generation of kinetic energy matches frictional loss sov remains constant as a satellite spirals inwards towards decreasing r. If n G r ∝ with 1 n > − as, for example, typically obtains close to the centers of galaxies FTNT24 , generation of kinetic energy falls short of matching frictional loss so v decreases as a satellite spirals inwards towards decreasing r.

Concluding Remarks
Introductory discussions were provided in Section 1. In Section 2 we discussed cyclostrophic flow, and derived the steepness and upper limit of the pressure gradient in vortices. In Section 3 we discussed the energy and power of vortices, including, in the case of atmospheric vortices, estimates of the number of times that the kinetic energy of a vortex must be regenerated during its lifetime to replace frictional dissipation. We explained 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. The effects on Earth's atmosphere as a whole of a cutoff of insolation, and of its partial cutoff in the winter hemisphere, were discussed. We considered only small atmospheric vortices, namely tornadoes, dust devils, waterspouts, and small hurricanes at low latitudes, so that we could assume that the Coriolis force can be neglected, and hence that the balance of forces on any parcel of moving air can be considered cyclostrophic [1] [2] with negligible error [1] [2]. [Waterspouts were given only limited consideration, because they are intermediate in size, intensity, and lifetime between dust devils and tornadoes, and hence their properties can be interpolated between those of dust devils and tornadoes. Even small hurricanes are much larger than even the largest tornadoes, let alone than even the largest waterspouts or dust devils, but they are still small enough that, especially at low latitudes, the Coriolis force can be neglected (except in their initial formative stages, which we did not consider).] We also considered whirlpools, which are even smaller and hence for which the cyclostrophic approximation [1] [2] is even more accurate [1] [2]. We neglected horizontal (constant-altitude) variations in fluid density ρ . 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. Indeed for whirlpools also neglecting vertical variations in water density ρ is an excellent approximation. Comparisons with geostrophic and friction-balanced flows were provided in Section 4, again where feasible neglecting variations in fluid density ρ . In Section 5 we considered an analogy that might be drawn, at least to some extent, with gravitational systems. We considered mainly spherically-symmetrical and cylindrically-symmetrical gravitational systems. In Section 6 we discussed generation of kinetic energy at the expense of potential energy in fluid vortices, in geostrophic and friction-balanced flows, and in gravitational systems. We ex-Open Journal of Fluid Dynamics plained the variations of pressure gradients and gravitational gradients corresponding to generation of kinetic energy equaling, exceeding, and falling short of frictional dissipation. Expanding on the second paragraph of Section 3.1 where windmills were briefly discussed, the Appendix describes a simple method for maximizing power extraction from environmental fluid (water or air) flows. It also briefly explains the application of this method to artificial (e.g., internal combustion) engines. Our overview of features and energetics of Earth's environmental fluid flows (focusing largely on vortices), even though mainly semiquantitative, hopefully may be helpful. Our attempt to draw analogies with respect to gravitational systems, even though of at most qualitative validity, hopefully may also be helpful. Despite the limitations of our analyses being mostly semiquantitative, hopefully they are helpful. Open Journal of Fluid Dynamics

Appendix: A Simple Method for Maximizing Power Extraction from Environmental Fluid Flows
In the second paragraph of Section 3.1, windmills were briefly discussed. In this Appendix, we expand on the second paragraph of Section 3.1, and describe a simple method for maximizing power extraction from environmental fluid (water or air) flows; e.g., power extraction from the flow of a river by a waterwheel, from the wind by a windmill, etc. If for example a waterwheel or windmill is spinning freely with no load imposed on it, so that it is not required to supply any torque  nor any power P , it will spin at its maximum possible angular velocity max ω in a given environmental fluid flow. (For a freely-spinning undershot waterwheel of radius  in a river flowing at linear velocity v, max v ω =  .) As the load imposed on a waterwheel or windmill is increased and it is required to supply increasing torque  , its angular velocity ω will decrease monotonically. At maximum possible load, with it being required to supply maximum possible torque max  , its angular velocity ω will have decreased to zero, so again it will supply zero power. Thus the power The Betz limit has been questioned for vertical-axis wind turbines [9] [10], but vertical-axis wind turbines generally have lower efficiencies than horizontal-axis ones [9] [10]. (But since they do not have to swivel into the wind, vertical-axis Open Journal of Fluid Dynamics wind turbines have fewer moving parts than horizontal-axis ones; also, they have balanced weight distributions about their centers and occupy less space than horizontal-axis wind turbines [9] [10].) An equivalent of the Betz limit for vertical-axis wind turbines might yet be derived [9] [10]. But whether or not the Betz limit is always an upper bound, a windmill will achieve its maximum possible It should be noted that novel systems for extracting energy from the wind are being developed. These include: (a) improved designs for vertical-axis wind turbines [30], (b) flying windmills [31], and (c) wind-harvesting systems with no moving parts [32] [33] [34]. FTN24 The latter share with vertical-axis wind turbines balanced weight distributions about their centers and occupying less space than horizontal-axis wind turbines-in addition to having no moving parts at all rather than merely fewer moving parts than horizontal-axis wind turbines. Perhaps Equations (A1) and (A2) could apply for nonrotary [32] [33] [34] wind-energy systems FTNT25 and nonrotary water-energy systems if appropriate analogs of  and ω were employed, and perhaps an equivalent of the Betz limit might (or might not) exist for nonrotary wind-energy and nonrotary water-energy systems. FTNT25 In extraction of power by rotary devices (e.g., waterwheels, horizontal-axis windmills, vertical-axis windmills, and flying windmills [30] [31]) from environmental fluid flows,  always decreases monotonically with increasing ω , because environmental fluid flows cannot increase with increasing ω of the power-extracting device. By contrast, for artificial engines (e.g., internal combustion engines), this need not be the case, because fuel flow to an artificial engine can increase with increasing ω of the engine: hence  can increase with increasing ω [35]. Nonetheless, fuel flow to an artificial (e.g., internal combustion) engine cannot increase with increasing indefinitely, and hence  cannot increase with increasing indefinitely [35]. Thus also for artificial (e.g., internal combustion) engines, (brake) power output is maximized as per the first three paragraphs of this Appendix [35]. (Note: In Figure 63  [35].)