_{1}

^{*}

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 <i>r</i><sub>eye</sub>. In the region <i>r</i><sub>eye</sub> ≤<i>r</i>≤<i>r</i><sub>max</sub> fluid (gas or liquid) circulates about the eye with speed <i>v</i> ∝<i>r</i><sup>n</sup>(<i>n</i><0). We take <i>r</i><sub>max</sub> 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 <i>n</i>=-1, angular momentum is conserved within the fluid itself; if <i>n</i> ≠-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.

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 r m a x towards decreasing radial distance r from the axis of rotation, i.e., v increasing monotonically with decreasing r, attaining a maximum value v m a x at the circumference of a calm area or eye of radius r eye 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 [

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 of this type, e.g., including Pacific typhoons, although we will occasionally refer to Pacific typhoons specifically.

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 [

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. Frictional losses typically result in − 1 < n < 0 . In rare cases generation of angular momentum and kinetic energy in vortices can exceed frictional losses, resulting in n < − 1 [A simple (nonvortex, noncyclonic) example for which n < − 1 : Let the speed v of rotation of a rigid hoop of radius 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”) n = 0 corresponds to constant v (v independent of r), and that n = + 1 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., ∂ P / ∂ r > 0 , in the region r eye ≤ r ≤ r m a x . Let the sea-level or ground-level pressure at the outer periphery of a vortex be P ( r m a x ) and that in the eye be P eye . Of course P eye < P ( r max ) . The pressure difference between r max and the eye is Δ P eye ≡ P ( r max ) − P eye > 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/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

If, as in the cases of most interest to us as per our construed definition of “vortex”, ^{FTNT0}; in a modified-Rankine-vortex model^{FTNT0} A calm eye and hence ^{FTNTS0,1}. More often than not there is little or no wind throughout the eye (not merely at the center of the eye) of atmospheric vortices, i.e.,

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.

Consider a small fluid parcel of mass m, volume V, density

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

(Sometimes

Strictly,

Now let

at

where

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

for even the strongest hurricanes and strongest tornadoes, a very good approximation for dust devils and waterspouts, and an excellent approximation for whirlpools. Thus in cases wherein ^{th}-centry work^{FINTIA} for in-depth explanations.) In typical hurricanes,

Let us briefly consider the range

Perhaps at this point, we should note that irrespective of the existence of eyes, all cyclones (including whirlpools) and all anticyclones must have calm areas at their centers, because their centers are minima and maxima, respectively, of pressure, so the pressure gradient G must vanish at their centers.^{FTNT1B} But an

eye implies centripetal force per unit mass of

that is no longer sufficient to impose further inflow to within

If, as in the cases of most interest to us as per our construed definition of “vortex”, ^{FTNT2}, if (corresponding to

where ^{ FTNT2}. Thus ^{ FTNT2}. Applying the last line of Equation (3) and Equation (6) yields

In the last term of Equation (7),

Letting^{FTNT2}, if (corresponding to

and

respectively, where ^{FTNT2}. Thus ^{FTNT2}. In Equations (8) and (9),

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

The solar constant at Earth is ≈1400 W/m^{2}. Over day and night, over all four 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,

A ^{FTNT3} that a maximum fraction ^{FTNT3}. [The dot-equal sign (^{FTNT3}. [There is some questioning of the Betz limit concerning vertical-axis windmills [^{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, ^{2} of Earth’s surface are required to supply each 1 m^{2} of such a windmill. Considering a flying windmill or kite windmill operating in the upper troposphere, at say 10^{4} m above Earth’s surface in middle latitudes, where ^{2} of Earth’s surface are required to supply each 1 m^{2} of such a windmill.

Consider first very small atmospheric vortices (tornadoes and dust devils). (Since waterspouts are intermediate in size, intensity, and lifetime between dust devils and tornadoes, we do not consider them explicitly, but interpolation between our results for dust devils and tornadoes can provide estimates.) The majority of the kinetic energy of these vortices is typically within the lower ≈1 km of Earth’s atmosphere. (Often their fastest winds are considerably closer to Earth’s surface than to ≈1 km above it.) Since we seek only approximate results we can take

be the surface area of Earth in the region

where

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

Hence during its lifetime the kinetic energy of a tornado’s or dust devil’s winds must be regenerated N times to replace frictional losses, where

In the third steps of Equations (15) and (16) we applied the approximation^{FTNT4}, tornado, fair estimates are

For a typical dust devil fair estimates are^{FTNT4A}.

Hence the kinetic energy of a typical tornado’s winds must be regenerated

typical dust devil’s winds must, essentially, be generated only once, the first time, because it does not live long enough for friction to dissipate a majority of the initially-generated kinetic energy of its winds. Of course, for exceptionally strong and/or long-lived tornadoes and dust devils, our estimates of N would be larger and for exceptionally weak and/or short-lived ones they would be smaller.^{FTNTS4,4A} Note that

Waterspouts are intermediate in size, intensity, and lifetime between dust devils and tornadoes, and hence values of corresponding quantities are also intermediate for waterspouts.

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 [

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

The power flux density of »2 W/m^{2} maintaining Earth’s winds against friction (recall the first two paragraphs of Section 3.1) sustains

We take the outer radius

be the surface area of Earth in the region

where

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

where

Hence during its lifetime a hurricane’s winds must be regenerated N times to replace frictional losses, where

In the third steps of Equations (22) and (23) we applied the approximation^{FTNT5}, small low-latitude hurricane fair estimates are

Thus the estimate of a typical hurricane’s energy as being about equal to that yielded by 2.2 megatons of TNT^{FTNTS6,7} or to that of a magnitude-7 earthquake^{FTNT7} (on the Richter scale) is a vast underestimate. Even a typical hurricane (

For Earth’s atmosphere as a whole, the root-mean-cube average wind speed is^{2} maintaining Earth’s winds against friction (recall the first two paragraphs of Section 3.1 and the third and fourth paragraphs of Section 3.2) sustains

winds contributing to ^{2} of Earth’s surface is »10^{4} kg. (Most of this atmospheric mass is, of course, within the troposphere.)

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 }

Hence ^{FTNT7A} for Earth’s troposphere as a whole is much longer than ^{2} of Earth’s surface, and also less friction per unit mass of air averaging over their entire depth, thus accounting for the longer

Thus if the supply of free energy ^{FTNT7A} (Of course

The quantity ^{FTNT8} It is free energy

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

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 ^{FTNT2}. Let

Thus the power available to a whirlpool is

Hence if the water is not replaced the e-folding time of a whirlpool is

The averages values in the denominators of Equation (30) obtain, approximately, when

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) remains applicable as it stands. Equations (2) and (3) are modified because the balance is now between the pressure-gradient force

where ^{FTNT2}, if frictional losses can be neglected,

where ^{FTNT2}. Thus ^{FTNT2}. Applying the last line of Equation (31) and Equation (32) yields

In the last two terms of Equation (33),

If^{FTNT2}, if frictional losses can be neglected, Equations (32) and (33) are obviously modified to

and

respectively. In Equations (34) and (35), ^{FTNT2}. Thus v is equal to the free-fall speed ^{FTNT2}. In the last two terms of Equation (35),

Corresponding to

Geostrophic (or quasi-geostrophic) flow is quite common from latitude

Let us also consider friction-balanced flows, wherein the pressure-gradient force per unit mass (or a component thereof) ^{FTNT1B} anticyclonic flows become increasingly difficult to maintain as ^{FTNT11} In such damming of cold waves, ^{FTNT11}: thus ^{FTNT11} At all typical wind and ocean-current speeds, ^{−1}, which accounts for, say, surface roughness. Applying Equation (1) and setting

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

Most typically,

Yet another example of friction-balanced flow is the flow of groundwater. The force driving the flow of groundwater, per unit mass, is^{−1}, which accounts for, say, the porosity and other properties of the materials comprising the water table that determine the ease or difficulty of groundwater flow. Thus^{FTNT13}:

Note that Darcy’s Law^{FTNT13} in hydrology [^{FTNT14} in electrical circuits [

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,

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

Can an analogy be drawn? A cute little book [^{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

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 [^{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 [^{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… [

For an isolated nonrotating spherically-symmetrical gravitator of radius ^{FTNT16}, i.e., for which General Relativity need not be employed for these purposes, applying Equations (4) and (5), at

[The universal gravitational constant

At ^{FTNT16} In Equations (39)-(43) and the associated discussions we choose ^{FTNT16} as per Bernoulli’s equation for fluid flow [^{FTNT2}: recall Equations (6)-(9) and the associated discussions. But we note that, applying Equation (2) and that the gravitational force between particles of masses M and m is

Strictly,

Radial spatial intervals ^{FTNTS17−17B}

For weak spherically-symmetrical gravitational fields at ^{FTNTS17−17B}

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 consistent 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 ^{FTNTS15A,17–17B}, but let us try anyway.

Can we draw the following analogy at

(more general geometry,)? (43)

The question marks in Equation (43) emphasize its speculative nature, and that it likely has at best only qualitative validity: For example: (a) In Equation (43) is it more correct to employ ^{FTNT18}.

Yet, at least prima facie, our result of Equation (43) seems to be qualitatively reasonable: that for a given

For the region of the Milky Way in the vicinity of the Sun, for most purposes Newtonian theory is sufficiently accurate. But this region is not in the space surrounding an isolated nonrotating spherically-symmetrical gravitator: the Milky Way rotates, is not perfectly spherically-symmetrical, and most importantly its mass is not entirely within the radius of the Sun’s orbit about the center of the Milky Way but extends well beyond the Sun’s orbit^{FTNTS19}^{-}^{21}. In the region of Milky Way in the vicinity of the Sun’s orbit, ^{FTNTS19}^{−}^{21}. [By Equations (39) and (40) and the associated discussions this implies that in the vicinity of the Sun’s orbit

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 [^{FTNT22}.

For comparison, let us consider (in the Newtonian approximation) the gravitational field of an isolated nonrotating long cylindrical mass M of radius

The gravitational force on a test particle of mass ^{FTNT23}):

In Equation (45)

Thus in this cylindrical case

Note the similarity of the results of Equations (44) and (47). Of course in this cylindrical case the test mass m is the only orbiting mass, whereas in the Milky Way in the vicinity of the Sun’s orbit there are numerous other orbiting masses, and these extend well beyond the Sun’s orbit. Thus note the similarity of the relation between the variation of G and the variation of v in these two cases, despite the difference in the physics between these two cases.

Calculation of

For an infinitely long (

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 [^{FTNT22}.

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 [^{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

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 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 [^{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

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

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

In cyclones with eyes—vortices as per our construed definition in the first paragraph of Section 1 (tornadoes, dust devils, waterspouts, hurricanes, and whirlpools)—maximum fluid speeds typically attain a large fraction of the maxima allowed by Bernoulli’s equation of energy conservation for fluid flow [^{FTNT2}. By contrast, in extratropical synoptic-scale weather systems (extratropical cyclones and anticyclones, and geostrophic flow) maximum wind speeds typically attain only a small fraction of the maxima thereby allowed, e.g., as per Equations (32) and (34) for geostrophic flow. This is in accordance with what is observed on typical weather maps. The pressure gradient typically steepens towards lower pressure much more closely to

In friction-balanced flow, most commonly, at least approximately, generation of kinetic energy matches frictional loss, so v remains at least approximately constant (in river flow v most typically increases downstream but only slightly). Hence in friction-balanced flow v usually does not attain a significant fraction of the maximum value allowed by energy conservation in accordance with Bernoulli’s equation of energy conservation for fluid flow [^{FTNT2}.^{ }

The first paragraph of this Section 6 applies, as per Section 5, in gravitational cases too. The pits in these cases are gravitational potential wells, but so too are, ultimately, the pits of pressure represented by cyclones and the bottoms of the hills of pressure represented by anticyclones and by geostrophic flows. In the absence of friction, a satellite orbits at fixed r and hence with fixed v. With friction it will spiral inwards and hence lose potential energy, which can be traded for a gain of kinetic energy and for frictional dissipation. Since, as per Equations (39) and (40) and the associated discussions, in the case of spherically-symmetrical gravitation ^{FTNT24}, generation of kinetic energy falls short of matching frictional loss so v decreases as a satellite spirals inwards towards decreasing r.

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 [

I am very grateful to Dr. Donald H. Kobe, Dr. Kurt W. Hess, and Dr. Stan Czamanski for very perceptive scientific discussions concerning fluid dynamics, especially those with Dr. Donald H. Kobe concerning fluid friction, those with Dr. Kurt W. Hess concerning tropical cyclones, windmills, and ocean currents, and those with Dr. Stan Czamanski pertinent to the Appendix. I thank Dr. Donald H. Kobe, Dr. Stan Czamanski, and Dr. S. Mort Zimmermanfor very insightful general scientific discussions over many years. I also thank Dr. Kurt W. Hess, Dan Zimmerman, and Robert H. Shelton for very insightful general scientific discussions at times. Additionally, I thank Robert H. Shelton for very helpful advice concerning diction.

Footnote 0: See also: (a) Ref. [

Footnote 1: See Ref. [

Footnote 1A: Reference [

Footnote 1B: There is an additional restriction on wind speed in anticyclones, which requires not only calms at their centers but upper limits on wind speed in general, if the flow is balanced. See Ref. 1, Section 7.2 (especially Subsection 7.2.6) and Ref. 2, Sections 1.1-1.3 and 3.2 (especially Subsection 3.2.5). If the flow is not balanced then this additional restriction with respect to upper limits on wind speed does not apply.

Footnote 2: In slightly generalized form Bernoulli’s equation of energy conservation for fluid flow can account for compressibility and frictional energy dissipation. See Refs. [

Footnote 3: See also: “Betz’s law” (most recently revised in 2019) at https://www.wikipedia.org, references cited therein, and other Wikipedia articles cited therein. Note: All Wikipedia articles have Talk pages, wherein strengths and weaknesses of the articles, along with suggestions for their improvement, are discussed.

Footnote 4: See: “Enhanced F Scale for Tornado Damage” (Update implemented on 1 February 2007) at https://www.spc.noaa.gov/faq/tornado/ef-scale.html. The distribution of lifetimes, sizes, and maximum wind speeds of tornadoes is very unsymmetrical: it is strongly positively skewed (see Weisstein, E.W. “Skewness.” From MathWorld— A Wolfram Web Resource. http://mathworld.wolfram.com/Skewness.html), with long tails extending towards high values well above the average. See for example: Ernest, A. and Childs, S. (2014) Adjustments in Tornado Counts, F-Scale Intensity, and Path Width for Assessing Significant Tornado Destruction. Journal of Applied Meteorology and Climatology, 53, 1494-1505; and Merritt, L. (2015) Tornado Frequency and Intensity in Oklahoma at http://apollo.ea.gatech.edu/EAS4480/2015/LaurenM Data Final Project.pptx.

Footnote 4A: The distribution of lifetimes, sizes, and maximum wind speeds of dust devils is very unsymmetrical: it is strongly positively skewed (see Weisstein, E.W. “Skewness” cited in Footnote 4) with long tails extending towards high values well above the average. This asymmetry is probably even more pronounced in the case of dust devils than in the case of tornadoes. Also in the case of dust devils there tends to be underestimation more than in the case of tornadoes, because the circulation of dust devils can extend beyond the range of visible dust. See for example: Sinclair, P.C. (1967) General Characteristics of Dust Devils. Journal of Applied Meteorology, 8, 32-45; and Cooley, J.R. (1971) Dust Devil Meteorology (NOAA Technical Memorandum NWSCR-42) at https://repository.library.noaa.gov/view/noaa/14125noaa_14125_DS1.pdf? (the short form of this website, noaa_14125_DS1.pdf, probably suffices for access).

Footnote 5: See: “Saffir-Simpson Hurricane Wind Scale” (Updated 2 January 2019 to include central North Pacific examples: left-click on “About the Saffir-Simpson Hurricane Wind Scale (PDF)” ) at https://www.nhc.noaa.gov/aboutsshws.php.

Footnote 6: See Ref. [

Footnote 7: See Ref. [

Footnote 7A: Our estimate of ≈1 week is comparable to that of ≈100 hours given in Subsection VI.11 “Energy Changes in Atmospheric Wind Systems” (see especially p. 471) of Stewart, H. J., Section VI “Kinematics and Dynamics of Fluid Flow”. In Berry, F.A., Jr., Bollay, E., and Beers, N.R., eds., Handbook of Meteorology, McGraw-Hill, New York, 1945.

Footnote 8: See Ref. [

Footnote 9: See Ref. [

Footnote 10: See Ref. [

Footnote 11: See Ref. [

Footnote 12: See Ref. [

Footnote 13: See also: “Darcy’s law” (most recently revised in 2019) at https://www.wikipedia.org, references cited therein, and other Wikipedia articles cited therein.

Footnote 14: See also Ref. [

Footnote 15: See Ref. [

Footnote 15A: See Ref. [

Footnote 15B: See also Ref. [

Footnote 16: See Ref. [

Footnote 17: Ruler distance is discussed, and distinguished from other distance measures in relativity, in Ref. [^{2} measures 2πr and 4πr^{2} even in the case of black holes: (i) for the Schwarzschild horizon with respect to black-hole dynamics (see Ref. [

Footnote 17A: The excess (extra-Euclidean) radial ruler distance (in the weak-field limit) of

Footnote 17B: See Ref. [

Footnote 18: See Ref. [

Footnote 19: See Ref. [

Footnote 20: See Ref. [

Footnote 21: See Ref. [

Footnote 22: See Ref. [

Footnote 23: See Ref. [

Footnote 24: See Ref. [

Footnote 25: See also: “Unconventional wind turbines” (most recently revised in 2019) at https://www.wikipedia.org, references cited therein, and other Wikipedia articles cited therein.

The author declares no conflicts of interest regarding the publication of 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

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

extracted from an environmental fluid flow will be maximized at intermediate values of

Let

Thus

positive slope

the optimum point (

Thus a waterwheel or a windmill will achieve its maximum possible power output ^{FTNT3} The Betz limit has been questioned for vertical-axis wind turbines [

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 [^{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 [^{FTNT25} and nonrotary water-energy systems if appropriate analogs of ^{FTNT25}

In extraction of power by rotary devices (e.g., waterwheels, horizontal-axis windmills, vertical-axis windmills, and flying windmills [