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Pressure Gradient, Power, and Energy of Vortices

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1. 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 ${r}_{max}$ towards decreasing radial distance r from the axis of rotation, i.e., $\partial v/\partial r<0$ (v increasing with decreasing r), attaining a maximum value ${v}_{max}$ at the circumference of a calm area or eye of radius ${r}_{\text{eye}}$ about the axis of rotation. (This monotonic increase $\partial v/\partial r<0$ of v with decreasing r in numerous instances of real vortices is interrupted by local fluctuations, but $\partial v/\partial r<0$ is the secular trend that we focus on.) Thus we construe tornadoes, waterspouts, dust devils, 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 rotating fluid (gas or liquid) systems will be in those meeting our construed definition of “vortex”, but we will also consider in some measure those not meeting this definition.

We consider small vortices, such as tornadoes, dust devils, small hurricanes at low latitudes, and whirlpools, for which the Coriolis force can be neglected. Waterspouts should also be mentioned. But because they 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, for 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, even though we occasionally mention Pacific typhoons specifically.

In these 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]. Such vortices are (at least approximately) cylindrically symmetrical about a vertical axis through the center of a calm central region or eye of radius ${r}_{\text{eye}}$. In the region ${r}_{\text{eye}}\le r\le {r}_{\mathrm{max}}$ fluid circulates about the eye with speed $v\propto {r}^{n}$ ; within the eye, at $0\le r<{r}_{\text{eye}}$, $v\approx 0\Rightarrow n\approx 0$. We take ${r}_{max}$ 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, not within our construed definition of vortex, $n\ge 0$ (of course $n>0$ cannot be maintained indefinitely with increasing r) then ${r}_{max}$ must be arbitrarily chosen.]

If $n=-1$, angular momentum is conserved within the fluid itself; if $n\ne -1$, angular momentum must be exchanged with Earth 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=-\infty $.] Note that $n=+1$ corresponds to solid-body (wheel-like) rotation, and that $n=0$ corresponds to constant v (v independent of r).

As we construe vortices to be cyclones with eyes, minimum pressure obtains in the eye, with pressure increasing monotonically with increasing r, i.e., $\partial P/\partial r>0$, in the region ${r}_{\text{eye}}\le r\le {r}_{max}$. Let the pressure at the outer periphery of a vortex be $P\left({r}_{max}\right)$ and that in the eye be ${P}_{\text{eye}}$. Of course ${P}_{\text{eye}}<P\left({r}_{\mathrm{max}}\right)$. The total pressure difference in the vortex is $\Delta {P}_{\text{total}}\equiv P\left({r}_{\mathrm{max}}\right)-{P}_{\text{eye}}>0$. For atmospheric vortices such as tornadoes, dust devils, waterspouts, and hurricanes, unless otherwise noted we take the fluid density $\rho $ to be that of air at or near sea level or ground level ( $\approx 1\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}$); for whirlpools we take $\rho $ to be density of water ( $\approx {10}^{3}\text{kg}/{\text{m}}^{\text{3}}$). We assume that horizontal (constant-altitude) changes in fluid density $\rho $ are small enough to neglect, i.e., that, corresponding to horizontal (constant-altitude) pressure differences $\Delta {P}_{\text{total}}\equiv P\left({r}_{\mathrm{max}}\right)-{P}_{\text{eye}}$, $\left|\Delta \rho \right|/\rho \ll 1$. 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. Indeed for whirlpools also neglecting vertical variations in water density $\rho $ is an excellent approximation. (All pressures and densities are at sea level or ground level unless otherwise noted.)

If, as in the cases of most interest to us as per our construed definition of “vortex”,
$n<0$ in the region
${r}_{\text{eye}}\le r\le {r}_{max}$, then fluid speed has its greatest value,
${v}_{max}=v\left({r}_{\text{eye}}\right)$ at
${r}_{\text{eye}}$, i.e., at the eye wall. In a Rankine-vortex model
$n=-1$ is assumed at
${r}_{\text{eye}}\le r\le {r}_{max}$ [3] [4] ;^{FTNT0} in a modified-Rankine-vortex model
$n<0$, usually but not necessarily restricted to within the range
$-1<n<0$, is assumed at
${r}_{\text{eye}}\le r\le {r}_{max}$ [3] [4].^{FTNT0} The assumption of a calm eye and hence
$v\approx 0\Rightarrow n\approx 0$ at
$0\le r<{r}_{\text{eye}}$ seems more consistent with and closer to observations than the assumption of solid-body (wheel-like) rotation within the eye and hence
$v={v}_{max}\left(r/{r}_{\text{eye}}\right)\Rightarrow n=+1$ at
$0\le r<{r}_{\text{eye}}$ as per Rankine-vortex [3] [4] or modified-Rankine-vortex models [3] [4] of tornadoes, dust devils, waterspouts, hurricanes, and whirlpools.^{FTNT0} Observations indicate that little fluid motion throughout the eye (not merely at the center of the eye), i.e.,
$v\approx 0\Rightarrow n\approx 0$ from
$r=0$ to very nearly
$r={r}_{\text{eye}}$, with v increasing from
$\approx 0$ to
${v}_{\mathrm{max}}=v\left({r}_{\text{eye}}\right)$ within a very short radial distance
$\ll {r}_{\text{eye}}$ just barely within
${r}_{\text{eye}}$, is more typical than
$v={v}_{max}\left(r/{r}_{\text{eye}}\right)\Rightarrow n=+1$ at
$0\le r<{r}_{\text{eye}}$.^{FTNT1} Indeed in the case of whirlpools there is not even any water within the eye at all, so
$v=0\Rightarrow n=0$ must strictly obtain within the entire range
$0\le r<{r}_{\text{eye}}$ !

In Section 2, we 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 are briefly discussed. Brief 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 briefly discussed in Section 6. We explain the variations of pressure and gravitational gradients corresponding to generation of kinetic energy exceeding, equaling, and falling short of frictional dissipation. Brief concluding remarks are given in Section 7. In the Appendix, we describe a simple method for maximizing power extraction from environmental fluid (air or water) flows. We briefly explain why this method is in general not applicable to artificial (e.g., internal combustion) engines. In summary, we provide an overview of features and energetics of Earth’s environmental fluid flows and of gravitational analogies thereto that, even though mostly semiquantitative, hopefully may be helpful.

2. Steepness and Upper Limit of Pressure Gradient in Vortices

Consider a small fluid parcel of mass m, volume V, density $\rho =m/V$, cross-sectional area A, and length L. Hence $V=AL=m/\rho $. Let the pressure be $P+\delta P$ one one side of the small fluid parcel and P on the other side at a distance L away. [Without loss of generality, we can take $\delta P\ge 0$. Since our fluid parcel is assumed small, $\delta P\ll \Delta {P}_{\text{total}}\equiv P\left({r}_{\mathrm{max}}\right)-{P}_{\text{eye}}$.] Thus the pressure gradient across our small fluid parcel is $G=\delta P/L$. The net pressure-gradient force ${F}_{\text{grad}}$ acting on our small fluid parcel is

${F}_{\text{grad}}=A\delta P=AL\frac{\delta P}{L}=VG=\frac{mG}{\rho}.$ (1)

If our small fluid parcel is moving on a circular path about the center of the 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

${F}_{\text{centr}}=\frac{m{v}^{2}}{r}.$ (2)

(Sometimes ${F}_{\text{centr}}$ is alternatively construed as the centrifugal force that the fluid parcel experiences and that balances the pressure-gradient force.) In cyclostrophic balance [1] [2]

${F}_{\text{centr}}={F}_{\text{grad}}$

$\Rightarrow \frac{m{v}^{2}}{r}=\frac{mG}{\rho}$

$\Rightarrow v={\left(\frac{Gr}{\rho}\right)}^{1/2}$

$\iff G=\frac{\rho {v}^{2}}{r}.$ (3)

Strictly, ${F}_{\text{grad}}$ 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

$v\propto {r}^{n}$ (4)

at ${r}_{\text{eye}}\le r\le {r}_{max}$. Applying Equation (4) to the last line of Equation (3) yields, at ${r}_{\text{eye}}\le r\le {r}_{max}$,

$\begin{array}{l}G\propto \frac{\rho \left(r\right)\times {r}^{2n}}{r}=\rho \left(r\right)\times {r}^{2n-1}\\ \approx {\langle \rho \left(r\right)\rangle}_{{r}_{\text{eye}}}^{{r}_{\mathrm{max}}}\times {r}^{2n-1},\end{array}$ (5)

where ${\langle \rho \left(r\right)\rangle}_{{r}_{\text{eye}}}^{{r}_{\mathrm{max}}}$ is the arithmetic average of $\rho \left(r\right)$ in the range ${r}_{\text{eye}}\le r\le {r}_{max}$

at the altitude where v and G are measured, most typically sea level or ground level. (Enclosure within angular brackets denotes the arithmetic average of the enclosed quantity.) Since $\partial \rho /\partial r<0$, i.e., since $\rho $ decreases radially inwards, the functional dependency of $\rho \left(r\right)$ on r, considered alone, results in G decreasing with decreasing r. The approximation in the second line of Equation (5), which neglects this functional dependency, is justified because, in accordance with the third-to-last paragraph of Section 1, corresponding to $\Delta {P}_{\text{total}}\equiv P\left({r}_{\mathrm{max}}\right)-{P}_{\text{eye}}$, $\left|\Delta \rho \right|/\rho \ll 1$ is even at worst a fairly good approximation for all vortices in Earth’s atmosphere, and always an excellent approximation for all whirlpools. Thus in cases where $n=-1$ is typically at least a close approximation―tornadoes, dust devils, waterspouts, and whirlpools―G increases very rapidly inwards, i.e., $G\propto {r}^{-3}$, in the region ${r}_{\text{eye}}\le r\le {r}_{max}$ [5]. (Reference [5] states this result but does not explain it.) In typical hurricanes, $-3/4\lesssim n\lesssim 1/2$, with the most typical value perhaps $n\approx 2/3$. Corresponding to $n=-3/4$, $G\propto {r}^{-5/2}$ ; corresponding to $n=-1/2$, $G\propto {r}^{-2}$ ; and corresponding to $n=-2/3$, $G\propto {r}^{-7/3}$.

Let us briefly consider the range $n\ge 0$ that is not within our construed definition of vortex, in light of Equation (5). Note that $n=0$, i.e., v independent of r, corresponds to $G\propto {r}^{-1}$ ; that $n=+1/2$ corresponds to $G\propto {r}^{0}$, i.e., to G independent of r; and that $n=+1$, i.e., solid-body (wheel-like) rotation, corresponds to $G\propto r$. Thus for solid-body (wheel-like) rotation―and only for solid-body (wheel-like) rotation―do both v and G vary identically with r (both directly proportional to r). (Of course $n>0$ cannot be maintained indefinitely with increasing r.)

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.^{ FTNT1A} But an eye

implies centripetal force ${\left[v\left({r}_{\text{eye}}\right)\right]}^{2}/{r}_{\text{eye}}={v}_{\mathrm{max}}^{2}/{r}_{\text{eye}}$ no longer sufficient to impose

further inflow to within ${r}_{\text{eye}}$ (sometimes alternatively construed as centrifugal force at ${r}_{\text{eye}}$ that prevents further inflow). Eyes exist only in vortices as per our construed definition in the first paragraph of Section 1: $n<0$ in the region ${r}_{\text{eye}}\le r\le {r}_{max}$ is a necessary (but not sufficient) condition for the existence of an eye. All eyes are calm areas at the centers of cyclones, but not vice versa. At the very least most extratropical cyclones lack eyes, and perhaps all do. All anticyclones lack eyes.

If, as in the cases of most interest to us as per our construed definition of “vortex”,
$n<0$ in the region
${r}_{\text{eye}}\le r\le {r}_{max}$, then the maximum fluid speed
${v}_{\mathrm{max}}=v\left({r}_{\text{eye}}\right)$ occurs at the eye wall, i.e., at
${r}_{\text{eye}}$, as does the maximum pressure gradient
${G}_{\mathrm{max}}=G\left({r}_{eye}\right)$. By Bernoulli’s equation of energy conservation for fluid flow [6] [7] ,^{FTNT2} if (corresponding to
$n=-1$) frictional losses can be neglected, then [6] [7]

${v}_{\mathrm{max}}=v\left({r}_{\text{eye}}\right)\approx {\left(\frac{2\Delta {P}_{\text{eye}}}{{\langle \rho \left(r\right)\rangle}_{{r}_{\text{eye}}}^{{r}_{\mathrm{max}}}}\right)}^{1/2}={\left(2g\Delta {H}_{\text{eye}}\right)}^{1/2},$ (6)

where
$\Delta {H}_{\text{eye}}$ is the dip of the isobaric surface that is at sea level or ground level in the eye from its altitude in the undisturbed atmosphere far from the vortex [6] [7].^{FTNT2} Thus
${v}_{\mathrm{max}}=v\left({r}_{\text{eye}}\right)$ is equal to the free-fall speed
${\left(2g\Delta {H}_{\text{eye}}\right)}^{1/2}$ from altitude
$\Delta {H}_{\text{eye}}$ [6] [7].^{FTNT2} Applying the last line of Equation (3) and Equation (6) yields

$\begin{array}{c}{G}_{\mathrm{max}}=G\left({r}_{\text{eye}}\right)\approx \frac{{\langle \rho \left(r\right)\rangle}_{{r}_{\text{eye}}}^{{r}_{\mathrm{max}}}{v}_{\mathrm{max}}^{2}}{{r}_{\text{eye}}}=\frac{{\langle \rho \left(r\right)\rangle}_{{r}_{\text{eye}}}^{{r}_{\mathrm{max}}}{\left[v\left({r}_{\text{eye}}\right)\right]}^{2}}{{r}_{\text{eye}}}\\ =\frac{2\Delta {P}_{\text{eye}}}{{r}_{\text{eye}}}=\frac{2{\langle \rho \rangle}_{\Delta {H}_{\text{eye}}}g\Delta {H}_{\text{eye}}}{{r}_{\text{eye}}}.\end{array}$ (7)

In the last term of Equation (7), ${\langle \rho \rangle}_{\Delta {H}_{\text{eye}}}$ is the average density of a vertical atmospheric column far from the vortex with base at sea level or ground level and top at $\Delta {H}_{\text{eye}}$.

Letting
$\Delta P\left({r}^{\prime}\right)\equiv P\left({r}_{max}\right)-P\left({r}^{\prime}\right)$, by Bernoulli’s equation of energy conservation for fluid flow [6] [7] ,^{FTNT2} if (corresponding to
$n=-1$) frictional losses can be neglected, we have similarly, at all
${r}^{\prime}$ in the region
${r}_{\text{eye}}\le {r}^{\prime}\le {r}_{max}$,

$v\left({r}^{\prime}\right)\approx {\left[\frac{2\Delta P\left({r}^{\prime}\right)}{{\langle \rho \left(r\right)\rangle}_{{r}^{\prime}}^{{r}_{max}}}\right]}^{1/2}\mathrm{=}{\left[2g\Delta H\left({r}^{\prime}\right)\right]}^{1/2}$ (8)

and

$G\left({r}^{\prime}\right)\approx \frac{{\langle \rho \left(r\right)\rangle}_{{r}^{\prime}}^{{r}_{\mathrm{max}}}{\left[v\left({r}^{\prime}\right)\right]}^{2}}{{r}^{\prime}}=\frac{2\Delta P\left({r}^{\prime}\right)}{{r}^{\prime}}=\frac{2{\langle \rho \rangle}_{\Delta H\left({r}^{\prime}\right)}g\Delta H\left({r}^{\prime}\right)}{{r}^{\prime}},$ (9)

where
$\Delta H\left({r}^{\prime}\right)$ is the dip of the isobaric surface that is at sea level or ground level at
${r}^{\prime}$ from its altitude in the undisturbed atmosphere far from the vortex [6] [7].^{FTNT2} Thus
$v\left({r}^{\prime}\right)$ is equal to the free-fall speed
${\left[2g\Delta H\left({r}^{\prime}\right)\right]}^{1/2}$ from altitude
$\Delta H\left({r}^{\prime}\right)$ [6] [7].^{FTNT2} In the second terms of Equations (8) and (9),
${\langle \rho \left(r\right)\rangle}_{{r}^{\prime}}^{{r}_{max}}$ is the arithmetic average of
$\rho \left(r\right)$ in the range
${r}^{\prime}\le r\le {r}_{max}$ at the altitude where v and G are measured, most typically sea level or ground level. [The approximations in the second steps of Equation (6) and (7) and in the first steps of Equations (8) and (9) are consistent with that in the second line of Equation (5), indeed, because
$\Delta P\left({r}^{\prime}\right)\le \Delta {P}_{\text{eye}}$, the latter two are better approximations than that in the second line of Equation (5).] In the last term of Equation (9),
${\langle \rho \rangle}_{\Delta H\left({r}^{\prime}\right)}$ is the average density of a vertical atmospheric column far from the vortex with base at sea level or ground level and top at
$\Delta H\left({r}^{\prime}\right)$. 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
$n=-1$. Owing to frictional losses, in actual vortices n is typically at least slightly larger than −1, and hence attainable values of
$v$ and
$G$ are typically at least 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
$n<-1$.)

3. Power and Energy of Vortices

3.1. Very Small Atmospheric Vortices: Tornadoes, Dust Devils, and Waterspouts

The solar constant at Earth is $\approx 1400\text{W}/{\text{m}}^{\text{2}}$. Over day and night, over all four seasons, and over clear and cloudy weather, the average solar flux absorbed (and thence reradiated) by Earth’s surface is $\approx 200\text{W}/{\text{m}}^{\text{2}}$. Of this $\approx 200\text{W}/{\text{m}}^{\text{2}}$, $\approx \mathrm{1\%}$ or $\approx 2\text{W}/{\text{m}}^{\text{2}}$ is converted into wind power. The power required to maintain wind speed v against friction is $\propto {v}^{3}$. A fair estimate of the root-mean-cube average surface wind speed ${v}_{\text{rmc,surface}}\equiv {\langle {v}^{3}\rangle}^{1/3}$ on Earth (at the official anemometer elevation of 10 m above Earth’s surface assuming no obstructions) is $\approx 5\text{m}/\text{s}$. Thus a fair estimate of the power flux density required to maintain surface wind speed v is

$\mathcal{P}\approx 2{\left(\frac{v}{{v}_{\text{rmc,surface}}}\right)}^{3}\text{W}/{\text{m}}^{\text{2}}\approx 2{\left(\frac{v}{5}\right)}^{3}\text{W}/{\text{m}}^{\text{2}}\approx \frac{{v}^{3}}{60}\text{W}/{\text{m}}^{\text{2}}\mathrm{.}$ (10)

A
${v}_{\text{rmc,surface}}\approx 5\text{m}/\text{s}$ wind at sea-level air density of
$\rho \approx 1\text{kg}/{\text{m}}^{\text{3}}$ delivers
$\mathcal{P}=\rho {v}_{\text{rmc,surface}}^{3}/2\approx 60\text{W}/{\text{m}}^{\text{2}}$ to a windmill. Thus
$\approx 30\text{\hspace{0.17em}}{\text{m}}^{2}$ 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. For a windmill at a higher elevation, say 200 m above Earth’s surface, where, say,
${v}_{\text{rmc,200m}}\approx 8\text{m}/\text{s}$,
$\mathcal{P}=\rho {v}_{\text{rmc,100m}}^{3}/2\approx 250\text{W}/{\text{m}}^{\text{2}}$ ; hence
$\approx 120\text{\hspace{0.17em}}{\text{m}}^{2}$ of Earth’s surface are required to supply each 1 m^{2} of such a windmill. For a flying windmill or kite windmill operating in the upper troposphere, at say 10^{4} m above Earth’s surface in middle

latitudes, ${\rho}_{{10}^{4}\text{m}}\approx \frac{1}{4}\text{kg}/{\text{m}}^{\text{3}}$ and say ${v}_{{\text{rmc,10}}^{\text{4}}\text{m}}\approx 30\text{m}/\text{s}$, $\mathcal{P}={\rho}_{{\text{10}}^{\text{4}}\text{\hspace{0.05em}}\text{m}}{v}_{{\text{rmc,10}}^{\text{4}}\text{\hspace{0.05em}}\text{m}}^{3}/2\approx 3400\text{W}/{\text{m}}^{\text{2}}$ ; hence $\approx 1700\text{\hspace{0.17em}}{\text{m}}^{2}$ of Earth’s surface are

required to supply each 1 m^{2} of such a windmill. It was proven by Albert Betz [8] [9] [10] that a maximum fraction of
$16/27\doteq 0.593=59.3\%$ of the wind’s energy can in principle be extracted by a windmill [8] [9] [10].^{FTNT3} [The dot-equal sign (
$\doteq $) means “very nearly equal to”.] Well-designed and well-built windmills can extract
$\approx \mathrm{45\%}$ of the wind’s energy, i.e.,
$\approx 3/4$ of the Betz limit [8] [9] [10].^{FTNT3} (There is some questioning of the Betz limit concerning vertical-axis wind turbines [9] [10].) A simple method for maximization of power extraction from environmental fluid (air or water) flows is discussed in the Appendix.

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.) These very small atmospheric vortices are typically confined to the lower $\approx 1\text{\hspace{0.17em}}\text{km}$ of Earth’s atmosphere but with their fastest winds typically considerably closer to Earth’s surface than to $\approx 1\text{\hspace{0.17em}}\text{km}$ above it. So since we seek only approximate results we can take ${v}_{\text{rmc,surface}}\approx 5\text{m}/\text{s}$ to be representative. We take the outer radius ${r}_{max}$ 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 obstructions) is $v=5\text{m}/\text{s}$. Thus we take $v\left({r}_{\mathrm{max}}\right)=5\text{m}/\text{s}$. Let

${S}_{\text{vortex}}=\text{\pi}\left({r}_{\mathrm{max}}^{2}-{r}_{\text{eye}}^{2}\right)\approx \text{\pi}{r}_{\mathrm{max}}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{r}_{\text{eye}}\ll {r}_{\mathrm{max}}$ (11)

be the surface area of Earth covered by the region within a tornado or dust devil between ${r}_{\text{eye}}$ and ${r}_{\mathrm{max}}$. Thus by Equations (10) and (11) the total power ${\mathbb{P}}_{\text{vortex}}=\mathcal{P}{S}_{\text{vortex}}$ required to maintain a tornado’s or dust devil’s wind against friction, and hence also the frictional dissipation, is

${\mathbb{P}}_{\text{vortex}}=\mathcal{P}{S}_{\text{vortex}}\approx \frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}}{60}\text{W},$ (12)

where ${v}_{\text{rms,vortex}}\equiv {\langle {v}_{\text{vortex}}^{3}\rangle}^{1/3}$ is the root-mean-cube average wind speed at

${r}_{\text{eye}}\le r\le {r}_{max}$ within the tornado or dust devil. The total energy required to maintain a tornado or dust devil for its lifetime ${\tau}_{\text{lifetime}}$, and ultimately frictionally dissipated, is

${E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)={\mathbb{P}}_{\text{vortex}}{\tau}_{\text{lifetime}}\approx \frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}{\tau}_{\text{lifetime}}}{60}\text{J}\mathrm{.}$ (13)

Since a tornado or dust devil is confined to the lower
$\approx 1\text{\hspace{0.17em}}\text{km}$ of Earth’s atmosphere, whose mass is » 1000 kg per m^{2} of Earth’s surface, its kinetic energy is

${E}_{\text{vortex,kin}}\approx \frac{1000{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}}{2}\text{J}=500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J},$ (14)

where ${v}_{\text{rms,vortex}}\equiv {\langle {v}_{\text{vortex}}^{2}\rangle}^{1/2}$ is the root-mean-square average wind speed at ${r}_{\text{eye}}\le r\le {r}_{max}$ 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

${\tau}_{\text{replacement}}\approx \frac{{E}_{\text{vortex,kin}}}{{\mathbb{P}}_{\text{vortex}}}\approx \frac{500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J}}{\frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}}{60}\text{W}}\approx \frac{3\times {10}^{4}}{\langle {v}_{\text{vortex}}\rangle}\text{s}.$ (15)

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

$N\approx \frac{{E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)}{{E}_{\text{vortex,kin}}}\approx \frac{\frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}{\tau}_{\text{lifetime}}}{60}\text{J}}{500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J}}\approx \frac{\langle {v}_{\text{vortex}}\rangle {\tau}_{\text{lifetime}}}{3\times {10}^{4}}\approx \frac{{\tau}_{\text{lifetime}}}{{\tau}_{\text{replacement}}}\mathrm{.}$ (16)

In the third steps of Equations (15) and (16) we apply the approximation
${v}_{\text{rmc,vortex}}^{3}\xf7{v}_{\text{rms,vortex}}^{2}\approx \langle {v}_{\text{vortex}}\rangle $, where
$\langle {v}_{\text{vortex}}\rangle $ is the arithmetic-average wind speed in the region
${r}_{\text{eye}}\le r\le {r}_{max}$ within a tornado or dust devil. Thus
${\tau}_{\text{replacement}}$ is approximately the e-folding time
${\tau}_{\text{efold}}$ for
$\langle {v}_{\text{vortex}}\rangle $ if the free-energy input that generates a tornado’s or dust devil’s winds and maintains its winds against friction were cut off. For a typical, say, EF3,^{FTNT4} tornado, fair estimates are
$\langle {v}_{\text{vortex}}\rangle \approx 40\text{m}/\text{s}$ and
${\tau}_{\text{lifetime}}\approx 2000\text{\hspace{0.17em}}\text{s}$ ; hence
${\tau}_{\text{replacement}}\approx 750\text{\hspace{0.17em}}\text{s}$

and $N\approx 2\frac{2}{3}$. For a typical dust devil fair estimates are $\langle {v}_{\text{vortex}}\rangle \approx 10\text{m}/\text{s}$ and

${\tau}_{\text{lifetime}}\approx 1000\text{\hspace{0.17em}}\text{s}$ ; hence ${\tau}_{\text{replacement}}\approx 3000\text{\hspace{0.17em}}\text{s}$ and $N\approx 0.3$. Hence the kinetic energy of a typical tornado’s winds must be regenerated $N\approx 2\frac{2}{3}$ times during its

lifetime to replace frictional dissipation. By contrast, a typical dust devil’s winds must, essentially, be generated only once, the first time, since 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. Note that ${\tau}_{\text{replacement}}$ and N are independent of ${S}_{\text{vortex}}$, because ${E}_{\text{vortex,kin}}/{\mathbb{P}}_{\text{vortex}}$ and ${E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)/{E}_{\text{vortex,kin}}$ are independent of ${S}_{\text{vortex}}$.

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

3.2. Larger (But Still Small) Atmospheric Vortices: Small Hurricanes

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, for 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 use the term “hurricanes” to encompass all tropical cyclones of this type, e.g., including Pacific typhoons (but 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 ${v}_{\text{rmc,trop}}\approx 10\text{m}/\text{s}$. Recalling the first two paragraphs of Section 3.1, the power flux required to maintain wind speed v within the lower half of the low-latitude troposphere is

$\mathcal{P}\approx 2{\left(\frac{v}{{v}_{\text{rmc}}}\right)}^{3}\text{W}/{\text{m}}^{\text{2}}\approx 2{\left(\frac{v}{10}\right)}^{3}\text{W}/{\text{m}}^{\text{2}}\mathrm{=}\frac{{v}^{3}}{500}\text{W}/{\text{m}}^{\text{2}}\mathrm{.}$ (17)

The power flux of $\approx 2\text{W}/{\text{m}}^{\text{2}}$ maintaining Earth’s winds against friction (recall the first two paragraphs of Section 3.1) sustains ${v}_{\text{rmc,trop}}\approx 10\text{m}/\text{s}\approx 2\times {v}_{\text{rmc,surface}}\approx 2\times 5\text{m}/\text{s}$ because the average friction over the entire lower half of the low-latitude troposphere is less than that at the official anemometer elevation for surface winds (10 m above Earth’s surface in the absence of obstructions). (The average wind speed is somewhat higher in the mid-latitude troposphere than the low-latitude troposphere owing to greater horizontal temperature contrasts in the former.)

We take the outer radius ${r}_{max}$ 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 ${v}_{\text{surface}}=5\text{m}/\text{s}$. Thus we take $v\left({r}_{\mathrm{max}}\right)=5\text{m}/\text{s}$. Let

${S}_{\text{vortex}}=\text{\pi}\left({r}_{\mathrm{max}}^{2}-{r}_{\text{eye}}^{2}\right)\approx \text{\pi}{r}_{\mathrm{max}}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{r}_{\text{eye}}\ll {r}_{\mathrm{max}}$ (18)

be the surface area of Earth covered by the region within a small low-latitude hurricane between ${r}_{\text{eye}}$ and ${r}_{max}$. Thus the total power ${\mathbb{P}}_{\text{vortex}}=\mathcal{P}{S}_{\text{vortex}}$ required to maintain the hurricane’s wind against friction, and hence also the frictional dissipation, is

${\mathbb{P}}_{\text{vortex}}=\mathcal{P}{S}_{\text{vortex}}\approx \frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}}{500}\text{W},$ (19)

where ${v}_{\text{rms,vortex}}\equiv {\langle {v}_{\text{vortex}}^{3}\rangle}^{1/3}$ is the root-mean-cube average wind speed at ${r}_{\text{eye}}\le r\le {r}_{max}$ within the hurricane. The total energy required to maintain a small low-latitude hurricane for its lifetime ${\tau}_{\text{lifetime}}$, and ultimately frictionally dissipated, is

${E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)={\mathbb{P}}_{\text{vortex}}{\tau}_{\text{lifetime}}\approx \frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}{\tau}_{\text{lifetime}}}{500}\text{J}.$ (20)

Since majority of the kinetic energy of the circulation of a small low-latitude hurricane is typically within the lower half of the atmosphere, whose mass is
$\approx 5\times {10}^{3}$ kg per m^{2} of Earth’s surface, its kinetic energy is

${E}_{\text{vortex,kin}}\approx \frac{5\times {10}^{3}{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}}{2}\text{J}=2500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J}\mathrm{,}$ (21)

where ${v}_{\text{rms,vortex}}\equiv {\langle {v}_{\text{vortex}}^{2}\rangle}^{1/2}$ is the root-mean-square average wind speed in the region ${r}_{\text{eye}}\le r\le {r}_{max}$ within the hurricane. Thus the kinetic energy of a hurricane’s winds must be replaced on a timescale

${\tau}_{\text{replacement}}\approx \frac{{E}_{\text{vortex,kin}}}{{\mathbb{P}}_{\text{vortex}}}\approx \frac{2500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J}}{\frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}}{500}\text{W}}\approx \frac{{10}^{6}}{\langle {v}_{\text{vortex}}\rangle}\text{s}\mathrm{.}$ (22)

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

$N\approx \frac{{E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)}{{E}_{\text{vortex,kin}}}\approx \frac{\frac{{S}_{\text{vortex}}{v}_{\text{rmc,vortex}}^{3}{\tau}_{\text{lifetime}}}{500}\text{J}}{2500{S}_{\text{vortex}}{v}_{\text{rms,vortex}}^{2}\text{J}}\approx {10}^{-6}\langle {v}_{\text{vortex}}\rangle {\tau}_{\text{lifetime}}\approx \frac{{\tau}_{\text{lifetime}}}{{\tau}_{\text{replacement}}}\mathrm{.}$ (23)

In the third steps of Equations (22) and (23) we apply the approximation
${v}_{\text{rmc,vortex}}^{3}\xf7{v}_{\text{rms,vortex}}^{2}\approx \langle {v}_{\text{vortex}}\rangle $, where
$\langle {v}_{\text{vortex}}\rangle $ is the arithmetic-average wind speed in the region
${r}_{\text{eye}}\le r\le {r}_{max}$ within a hurricane. Thus
${\tau}_{\text{replacement}}$ is approximately the e-folding time
${\tau}_{\text{efold}}$ for
$\langle {v}_{\text{vortex}}\rangle $ if the free-energy input that generates a hurricane’s winds and maintains its winds against friction were cut off. For a typical, say, Category 3, ^{FTNT5} small low-latitude hurricane fair estimates are
$\langle {v}_{\text{vortex}}\rangle \approx 30\text{m}/\text{s}$ and
${\tau}_{\text{lifetime}}\approx 2\text{\hspace{0.17em}}\text{weeks}\approx 1.2\times {10}^{6}\text{s}$ ; hence

${\tau}_{\text{replacement}}\approx 3\times {10}^{4}\text{\hspace{0.17em}}\text{s}\approx \frac{1}{3}\text{\hspace{0.17em}}\text{d}$ and $N\approx 40$. Hence the kinetic energy of a typical

small low-latitude hurricane’s winds must be regenerated $\approx 40$ times during its lifetime to replace frictional dissipation. Note that, for hurricanes as for tornadoes, dust devils, and waterspouts, ${\tau}_{\text{replacement}}$ and N are independent of ${S}_{\text{vortex}}$, because ${E}_{\text{vortex,kin}}/{\mathbb{P}}_{\text{vortex}}$ and ${E}_{\text{vortex}}\left({\tau}_{\text{lifetime}}\right)/{E}_{\text{vortex,kin}}$ are independent of ${S}_{\text{vortex}}$.

Thus the estimate of a typical hurricane’s energy as being about equal to that yielded by 2.2 megatons of TNT^{FTNT6,7} or to that of a magnitude-7 earthquake^{FTNT7} (on the Richter scale) is a vast underestimate. A typical small low-latitude hurricane generates, and ultimately frictionally dissipates during its lifetime, kinetic energy
$\approx 40$ times that yielded by 2.2 megatons of TNT or
$\approx 40$ times that of a magnitude-7 earthquake (on the Richter scale). Even this figure is exceeded by exceptionally intense, large, and/or long-lived Atlantic, Caribbean, and Gulf-of-Mexico hurricanes, and is even more strongly exceeded by exceptionally intense, large, 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.] By contrast, even the most active earthquake faults are doing very well to manage one magnitude-7 earthquake per timescale of decades [11] , and one magnitude-7 earthquake per timescale of centuries or millennia is more typical [11].

3.3. 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 ${v}_{\text{rmc,Earth}}\approx 15\text{m}/\text{s}$. The power flux of $\approx 2\text{W}/{\text{m}}^{\text{2}}$ 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

${v}_{\text{rmc,Earth}}\approx 15\text{m}/\text{s}\approx 3\times {v}_{\text{rmc,surface}}\approx 3\times 5\text{m}/\text{s}\approx 1\frac{1}{2}\times {v}_{\text{rmc,trop}}\approx 1\frac{1}{2}\times 10\text{m}/\text{s}$ because the

fastest winds contributing to
${v}_{\text{rmc,Earth}}\approx 15\text{m}/\text{s}$ are in the upper troposphere, where there is much less friction than at the official anemometer elevation for surface winds (10 m above Earth’s surface assuming no obstructions), with intermediate friction averaging over the lower half of the troposphere. The mass of Earth’s atmosphere per m^{2} of Earth’s surface is
$\approx {10}^{4}\text{kg}$. (Most of this atmospheric mass is, of course, within the troposphere.)

Neglecting the difference between root-mean-cube and root-mean-square average wind speeds, the total kinetic energy of wind per m^{2} of Earth’s surface is

${E}_{\text{kin}}/{\text{m}}^{2}\approx \frac{1}{2}\times {10}^{4}\times {15}^{2}\text{J}/{\text{m}}^{\text{2}}\approx {10}^{6}\text{J}/{\text{m}}^{\text{2}}.$ (24)

Thus the replacement timescale for the kinetic energy of the Earth’s winds is

${\tau}_{\text{replacement}}\approx \frac{{10}^{6}\text{J}/{\text{m}}^{\text{2}}}{2\text{W}/{\text{m}}^{\text{2}}}=5\times {10}^{5}\text{\hspace{0.05em}}\text{s}\approx 1\text{\hspace{0.17em}}\text{week}.$ (25)

Hence ${\tau}_{\text{replacement}}\approx 5\times {10}^{5}\text{s}\approx 1\text{\hspace{0.17em}}\text{week}$ for Earth’s troposphere as a whole is much longer than ${\tau}_{\text{replacement}}\approx 750\text{\hspace{0.17em}}\text{s}$ for tornadoes, ${\tau}_{\text{replacement}}\approx 3000\text{\hspace{0.17em}}\text{s}$ for dust

devils, and even ${\tau}_{\text{replacement}}\approx 3\times {10}^{4}\text{\hspace{0.17em}}\text{s}\approx \frac{1}{3}\text{\hspace{0.17em}}\text{d}$ for hurricanes. In tornadoes,

${\tau}_{\text{replacement}}$ is very short because very high wind speeds occur near the ground, where friction is greatest. In dust devils, maximum winds also occur near the ground but they are much slower than in tornadoes, so
${\tau}_{\text{replacement}}$ is longer for dust devils than for tornadoes. (Waterspouts are intermediate between tornadoes and dust devils.) In hurricanes wind speeds are high, much closer to those in tornadoes than to those in dust devils, but hurricanes are much deeper vertically than dust devils or even tornadoes, so there is much more kinetic energy to be frictionally dissipated per m^{2} of Earth’s surface, which largely accounts for the longer
${\tau}_{\text{replacement}}$ for hurricanes. But for Earth’s atmosphere as a whole the vertical depth is maximized at the entire depth of the atmosphere (with sufficient accuracy at the entire depth of the troposphere), and maximum wind speeds are confined to the highest altitudes of the troposphere where friction is least. Hence for Earth’s atmosphere as a whole
${\tau}_{\text{replacement}}$ is much longer than even the longest of the other three given values (for hurricanes).

Thus if the supply of free energy $\mathbb{F}$ to Earth’s atmosphere were cut off, Earth’s winds would cease with an e-folding time ${\tau}_{\text{efold}}$ for $\langle v\rangle $ of approximately one week. But if the Sun were turned off, the supply of free energy $\mathbb{F}$ to Earth’s atmosphere would not be immediately cut off: Earth’s winds would not cease with an e-folding time ${\tau}_{\text{efold}}$ for $\langle v\rangle $ of approximately one week, because the very large amount of thermal energy ${E}_{\text{ocean}}$ stored in Earth’s oceans would then become much more strongly thermodynamically available. With insolation cut off, the continents would cool much faster (initially say at $\approx 40{}^{\circ}\text{F}/\text{d}\approx 20{}^{\circ}\text{C}/\text{d}$)

than the oceans (initially say at $\approx 1{}^{\circ}\text{F}/\text{d}\approx \frac{1}{2}{}^{\circ}\text{C}/\text{d}$), so very large temperature

differences would be generated between the temperature ${T}_{\text{cont}}$ of the very cold continents and the surface temperature ${T}_{\text{ocean}}$ of the much less cold oceans. Thus a greatly increased fraction of the thermal energy ${E}_{\text{ocean}}$ stored in Earth’s oceans would be upgraded to free energy $\mathbb{F}$ :

$\mathbb{F}={E}_{\text{ocean}}\left(1-\frac{{T}_{\text{cont}}}{{T}_{\text{ocean}}}\right)\mathrm{.}$ (26)

The quantity
$1-\frac{{T}_{\text{cont}}}{{T}_{\text{ocean}}}$ is of course the Carnot efficiency.^{FTNT8} It is free energy

$\mathbb{F}$ that generates Earth’s winds and maintains them in the face of frictional losses. If the Sun were turned off, ${E}_{\text{ocean}}$ would decrease only very slowly, but

$1-\frac{{T}_{\text{cont}}}{{T}_{\text{ocean}}}$ would initially increase very rapidly, so the free energy $\mathbb{F}$ available

to Earth’s atmosphere would initially be greatly increased. Hence wind speeds would initially increase: a period of strong winds would ensue, especially along and near the coasts of the continents, where the temperature gradient between the continents and the oceans would be steepest. As the oceans froze over,

$1-\frac{{T}_{\text{cont}}}{{T}_{\text{ocean}}}$ and thus $\mathbb{F}$, and hence Earth’s winds, would gradually diminish. Yet

even through a layer of ice say ~1 m thick enough heat would flow from the liquid ocean below to keep ${T}_{\text{ocean}}$ (now the surface temperature of the oceanic ice layer) considerably warmer than ${T}_{\text{cont}}$. Only after the oceans had frozen over

to a sufficient depth, which would probably require a timescale of months,

would $1-\frac{{T}_{\text{cont}}}{{T}_{\text{ocean}}}$ and thus $\mathbb{F}$ finally vanish, and hence Earth’s winds finally cease.

3.4. 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

${v}_{\mathrm{max}}={\left(\frac{2\Delta {P}_{\text{eye}}}{\rho}\right)}^{1/2}={\left(\frac{2\rho gH}{\rho}\right)}^{1/2}={\left(2gH\right)}^{1/2},$ (27)

where
$\rho \approx {10}^{3}\text{kg}/{\text{m}}^{\text{3}}$ 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 [6] [7] , and
${\left(2gH\right)}^{1/2}$ is the free-fall speed from
$H$ (similarly as is the case with atmospheric vortices [6] [7] ).^{FTNT2} Let
$\mathfrak{M}$ be the total mass of water in the sink. If
${S}_{\text{sink}}$, the surface area of the sink, is large compared to the area where the water level is significantly depressed by the whirlpool, the center of mass of the water in the sink
${H}_{\text{cm}}$ is at elevation
$\approx H/2$ above the floor of the sink. Then
$\mathfrak{M}=2\rho {S}_{\text{sink}}{H}_{\text{cm}}\approx \rho {S}_{\text{sink}}H,$ and the total gravitational potential energy of the water relative to the floor of the sink is

$\begin{array}{l}{E}_{\text{pot}}=\mathfrak{M}g{H}_{\text{cm}}=\left(2\rho {S}_{\text{sink}}{H}_{\text{cm}}\right)g{H}_{\text{cm}}=2\rho g{S}_{\text{sink}}{H}_{\text{cm}}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx \frac{1}{2}\mathfrak{M}gH\approx \frac{1}{2}\left(\rho {S}_{\text{sink}}H\right)gH=\frac{1}{2}\rho g{S}_{\text{sink}}{H}^{2}.\end{array}$ (28)

Thus the power available to the whirlpool is

$\begin{array}{l}{\mathbb{P}}_{\text{vortex}}=\left|\frac{\text{d}{E}_{\text{pot}}}{\text{d}t}\right|=g\left|\frac{\text{d}\left(\mathfrak{M}{H}_{\text{cm}}\right)}{\text{d}t}\right|=2\rho g{S}_{\text{sink}}\left|\frac{\text{d}\left({H}_{\text{cm}}^{2}\right)}{\text{d}t}\right|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx \frac{1}{2}g\left|\frac{\text{d}\left(\mathfrak{M}H\right)}{\text{d}t}\right|=\frac{1}{2}\rho g{S}_{\text{sink}}\left|\frac{\text{d}\left({H}^{2}\right)}{\text{d}t}\right|.\end{array}$ (29)

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

$\begin{array}{l}{\tau}_{\text{efold}}\approx \frac{{E}_{\text{pot}}}{\langle {\mathbb{P}}_{\text{vortex}}\rangle}\approx \frac{{H}_{\text{cm}}^{2}}{\langle \left|\text{d(}{H}_{\text{cm}}^{2})/\text{d}t\right|\rangle}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{{H}^{2}}{\langle \left|\text{d}({H}^{2})/\text{d}t\right|\rangle}.\end{array}$ (30)

The average values in the denominators of Equation (30) obtain, approximately, when ${H}_{\text{cm}}^{2}$ and therefore also ${H}^{2}\approx 4{H}_{\text{cm}}^{2}$ have been reduced to 1/2 of their initial values, or, equivalently, when ${H}_{\text{cm}}$ and therefore also $H\approx 2{H}_{\text{cm}}$ have been reduced to $1/{2}^{1/2}$ of their initial values.

4. Geostrophic and Friction-Balanced Flows

Although our main concern in this paper is with cyclostrophic flow, brief comparisons with geostrophic flow (straight isobars),^{FTNT9,10} and with friction-balanced flow, 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 $mG/\rho $ and the Coriolis force $2mv\Omega sin\varphi $ as per

${F}_{\text{Coriolis}}={F}_{\text{grad}}$

$\Rightarrow 2mv\Omega \mathrm{sin}\varphi =\frac{mG}{\rho}$

$\Rightarrow v=\frac{G}{2\rho \Omega \mathrm{sin}\varphi}$

$\iff G=2\rho v\Omega \mathrm{sin}\varphi ,$ (31)

where
$\Omega =2\text{\pi}\text{rad}/\text{d}\doteq 7.292\times {10}^{-5}\text{rad}/\text{s}$ is Earth’s angular sidereal rotational speed and
$\varphi $ is the latitude. Let
$\Delta {P}_{\text{total}}\equiv {P}_{\mathrm{max}}-{P}_{\mathrm{min}}$ be the total pressure difference between maximum and minimum surface barometric pressure in geostrophic flow. Then by Bernoulli’s equation of energy conservation for fluid flow [6] [7] ,^{FTNT2} if frictional losses can be neglected,

$\begin{array}{l}{v}_{\mathrm{max}}\approx {\left(\frac{2\Delta {P}_{\text{total}}}{{\langle \rho \rangle}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}}\right)}^{1/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(2g\Delta {H}_{\mathrm{max}}\right)}^{1/2},\end{array}$ (32)

where ${\langle \rho \rangle}_{{P}_{min}}^{{P}_{max}}$ is the arithmetic average of $\rho $ over the range of pressures

${P}_{min}\le P\le {P}_{max}$ at the altitude where v and G are measured, most typically 10 m above the surface and sea level or ground level, respectively, and
$\Delta {H}_{max}$ is the dip of the isobaric surface that is at sea level or ground level at the locations of minimum surface barometric pressure
${P}_{min}$ from its altitude at the locations of maximum surface barometric pressure
${P}_{max}$ [6] [7].^{FTNT2} Thus
${v}_{max}$ is equal to the free-fall speed
${\left(2g\Delta {H}_{max}\right)}^{1/2}$ from altitude
$\Delta {H}_{max}$ [6] [7].^{FTNT2} Applying the last line of Equation (31) and Equation (32) yields

$\begin{array}{c}{G}_{\mathrm{max}}\approx 2{\langle \rho \rangle}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}{v}_{\mathrm{max}}\Omega \mathrm{sin}\varphi =2{\langle \rho \rangle}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}{\left(\frac{2\Delta {P}_{total}}{{\langle \rho \rangle}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}}\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ ={2}^{3/2}{\left({\langle \rho \rangle}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}\Delta {P}_{\text{total}}\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ =2{\langle \rho \rangle}_{\Delta {H}_{\mathrm{max}}}{\left(2g\Delta {H}_{\mathrm{max}}\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ ={2}^{3/2}{\langle \rho \rangle}_{\Delta {H}_{\mathrm{max}}}{\left(g\Delta {H}_{\mathrm{max}}\right)}^{1/2}\Omega \mathrm{sin}\varphi .\end{array}$ (33)

In the last two terms of Equation (33), ${\langle \rho \rangle}_{\Delta {H}_{max}}$ is the average density of a vertical atmospheric column where the surface pressure is ${P}_{max}$ with base at sea level or ground level and top at $\Delta {H}_{max}$.

At locations where
${P}_{\mathrm{min}}<{P}^{\prime}<{P}_{\mathrm{max}}$, letting
$\Delta P\equiv {P}^{\prime}-{P}_{min}$, by Bernoulli’s equation of energy conservation for fluid flow [6] [7] ,^{FTNT2} if frictional losses can be neglected, Equations (32) and (33) are obviously modified to

$v\approx {\left(\frac{2\Delta P}{{\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{\mathrm{max}}}}\right)}^{1/2}={\left(2g\Delta {H}^{\prime}\right)}^{1/2}$ (34)

and

$\begin{array}{c}G\approx 2{\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{\mathrm{max}}}v\Omega \mathrm{sin}\varphi =2{\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{\mathrm{max}}}{\left(\frac{2\Delta P}{{\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{\mathrm{max}}}}\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ ={2}^{3/2}{\left({\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{\mathrm{max}}}\Delta P\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ =2{\langle \rho \rangle}_{\Delta {H}^{\prime}}{\left(2g\Delta {H}^{\prime}\right)}^{1/2}\Omega \mathrm{sin}\varphi \\ ={2}^{3/2}{\langle \rho \rangle}_{\Delta {H}^{\prime}}{\left(g\Delta {H}^{\prime}\right)}^{1/2}\Omega \mathrm{sin}\varphi ,\end{array}$ (35)

respectively. In Equations (34) and (35),
${\langle \rho \rangle}_{{P}^{\prime}}^{{P}_{max}}$ is the arithmetic average of
$\rho $ over the range of pressures
${P}^{\prime}\le P\le {P}_{max}$ at the altitude where v and G are measured, most typically 10 m above the surface and sea level or ground level, respectively, and
$\Delta {H}^{\prime}$ is the dip of the isobaric surface that is at sea level or ground level at the locations of surface barometric pressure
${P}^{\prime}$ from its altitude at the locations of maximum surface barometric pressure
${P}_{max}$ [6] [7].^{FTNT2} Thus v is equal to the free-fall speed
${\left(2g\Delta {H}^{\prime}\right)}^{1/2}$ from altitude
$\Delta {H}^{\prime}$ [6] [7].^{FTNT2} In the last two terms of Equation (35),
${\langle \rho \rangle}_{\Delta {H}^{\prime}}$ is the average density of a vertical atmospheric column where the surface pressure is
${P}_{max}$ with base at sea level or ground level and top at
$\Delta {H}^{\prime}$. Again, Equations (31)-(35) represent theoretical upper limits, neglecting frictional losses.

Corresponding to $\Delta {P}_{\text{total}}\equiv {P}_{max}-{P}_{min}$, and hence even more so corresponding to $\Delta P\equiv {P}^{\prime}-{P}_{min}$, $\left|\Delta \rho \right|/\rho \ll 1$ 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 (quasi-geostrophic) flows in Earth’s oceans. Thus the approximations in the first steps of Equations (32)-(35) are consistent with those in the second line of Equation (5), the second steps of Equations (6) and (7), and the first steps of Equations (8) and (9). Geostrophic (or quasi-geostrophic) flow is quite common from approximately latitude $\varphi \approx {20}^{\circ}$ polewards, both in Earth’s atmosphere and in its oceans. Of course, in Earth’s lower atmosphere, $\rho \approx 1\text{kg}/{\text{m}}^{\text{3}}$, whereas in its oceans, $\rho \approx {10}^{3}\text{kg}/{\text{m}}^{\text{3}}$.

Let us also consider friction-balanced flow, wherein the pressure-gradient force per unit mass (or a component thereof)
${F}_{\text{grad}}/m$ balances the frictional force per unit mass
${F}_{\text{fric}}/m$. Friction-balanced flow often occurs at low latitudes, say
$\varphi \lesssim {20}^{\circ}$, especially within a few degrees of or at the equator, where the Coriolis force
${F}_{\text{Coriolis}}$ is small (zero at the equator). Geostrophic flow thus becomes increasingly difficult to maintain as
$\varphi $ decreases below
$\approx {20}^{\circ}$, and is impossible to maintain at the equator. In approaching the equator from latitude
$\varphi \approx {20}^{\circ}$, flow under straight (or quasi-straight) isobars becomes less geostrophic (or quasi-geostrophic) and more friction-balanced, until at the equator it must be purely friction-balanced. Friction-balanced flow can also occur even at high latitudes in, for example, the damming of cold waves by mountain ranges.^{FTNT11} In such damming of cold waves,
${F}_{\text{Coriolis}}$ is balanced by the force exerted on the cold air by a mountain range, and typically only a component of the pressure-gradient force balances
${F}_{\text{fric}}$ :^{FTNT11} thus
${F}_{\text{grad}}$ should in such cases be construed as the component of the pressure-gradient force that balances
${F}_{\text{fric}}$.^{FTNT11} Within the entire range of wind speeds on Earth,
${F}_{\text{fric}}/m$ is, at least approximately, proportional to
${v}^{2}$, i.e.,
${F}_{\text{fric}}/m\approx \mathcal{C}{v}^{2}$, where
$\mathcal{C}$ is a constant. Applying Equation (1) and setting
${F}_{\text{fric}}/m={F}_{\text{grad}}/m$ yields

$\frac{{F}_{\text{fric}}}{m}\approx \mathcal{C}{v}^{2}=\frac{{F}_{\text{grad}}}{m}=\frac{G}{\rho}$

$\Rightarrow v\approx {\left(\frac{G}{\mathcal{C}\rho}\right)}^{1/2}\mathrm{.}$ (36)

Another example of friction-balanced flow is a river in steady state.^{FTNT12} The force driving the flow of a river, per unit mass of flowing water, is
${F}_{\text{driv}}/m=g\mathrm{sin}\theta $, where g is the acceleration due to gravity and
$\theta $ is the slope. In almost all cases
$\theta \ll 1\text{\hspace{0.17em}}\text{rad}$ so
$sin\theta \doteq \theta $ and hence
${F}_{\text{driv}}\doteq g\theta $. The frictional force retarding the motion of water in a river flowing at speed v, per unit mass of flowing water, within the entire range of river-flow speeds on Earth, is, at least approximately, proportional to
${v}^{2}$. Because most of the friction retarding the flow of a river occurs via interaction with its river bed, this retarding frictional force, per unit mass of flowing water, is also, at least approximately, proportional to
$\mathcal{P}/\mathcal{A}$, where
$\mathcal{P}$ is the river’s wetted perimeter and
$\mathcal{A}$ is its cross-sectional area. Thus
${F}_{\text{fric}}/m\approx {\mathcal{C}}^{\prime}\mathcal{P}{v}^{2}/\mathcal{A}$, where
${\mathcal{C}}^{\prime}$ is a constant. Similarly to Equation (36), setting
${F}_{\text{fric}}/m={F}_{\text{driv}}/m$ yields

$\begin{array}{l}\frac{{F}_{\text{fric}}}{m}\approx \frac{{\mathcal{C}}^{\prime}\mathcal{P}{v}^{2}}{\mathcal{A}}=\frac{{F}_{\text{driv}}}{m}=gsin\theta \stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta \\ \Rightarrow v\approx {\left(\frac{\mathcal{A}gsin\theta}{{\mathcal{C}}^{\prime}\mathcal{P}}\right)}^{1/2}\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}{\left(\frac{\mathcal{A}g\theta}{{\mathcal{C}}^{\prime}\mathcal{P}}\right)}^{1/2}\mathrm{.}\end{array}$ (37)

In all rivers on Earth g can of course be considered constant for all practical

purposes. Downstream in most rivers on Earth in steady state $sin\theta \stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}\theta $

decreases ^{FTNT12} but
$\mathcal{A}/\mathcal{P}$ increases. In most rivers on Earth in steady state
$\mathcal{A}/\mathcal{P}$ increases downstream slightly faster than
$sin\theta \stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}\theta $ decreases, so v increases slightly downstream.

Yet another example of friction-balanced flow is the steady-state flow of ground water. The force driving the flow of ground water, per unit mass, is
${F}_{\text{driv}}/m=gsin\theta $, where g is the acceleration due to gravity and
$\theta $ is the hydraulic gradient, i.e., the slope of the water table. In almost all cases
$\theta \ll 1\text{\hspace{0.17em}}\text{rad}$ so
$sin\theta \doteq \theta $ and hence
${F}_{\text{driv}}\doteq g\theta $. The frictional force retarding the motion of ground water flowing at speed v, per unit mass, within the entire range of ground-water-flow speeds on Earth, is, at least approximately, proportional to v. Thus
${F}_{\text{fric}}/m\approx {\mathcal{C}}^{\u2033}v$, where
${\mathcal{C}}^{\u2033}$ is a constant. (
${\mathcal{C}}^{\u2033}$ usually decreases with increasing porosity of the materials comprising the water table, through which ground water flows.) Similarly to Equations (36) and (37), setting
${F}_{\text{fric}}/m={F}_{\text{driv}}/m$ yields Darcy’s Law [12] [13] : ^{FTNT13}

$\frac{{F}_{\text{fric}}}{m}\approx {\mathcal{C}}^{\u2033}v=\frac{{F}_{\text{driv}}}{m}=g\mathrm{sin}\theta \stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta $

$\Rightarrow v\approx \frac{g\mathrm{sin}\theta}{{\mathcal{C}}^{\u2033}}\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}\frac{g\theta}{{\mathcal{C}}^{\u2033}}.$ (38)

Note that Darcy’s Law in hydrology [12] [13] is essentially equivalent to Ohm’s law ^{FTNT14} in electrical circuits [13].

Ground-water flows occur at much smaller Reynolds numbers than river-water flows or atmospheric winds. At small Reynolds numbers (as for ground-water flows, with rare exceptions^{FTNT13}) viscous drag is predominant so, at least approximately,
${F}_{\text{fric}}\propto v$ : viscous shear forces are, at least to a first approximation, proportional to v [14] [15]. At large Reynolds numbers (as for river-water flows and atmospheric winds), inertial drag is predominant, so, at least to a first approximation,
${F}_{\text{fric}}\propto {v}^{2}$ : both (i) the mass of fluid and (ii) the velocity change and hence momentum change imparted per mass of fluid moving turbulently past a given location are, at least to a first approximation, proportional to v; two factors of v amount to
${v}^{2}$ [14] [15].

5. Gravity

5.1. Spherically-Symmetrical Gravity

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 and dust, 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 ${F}_{\text{grad}}$ 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 the present paper. Such an analogy may also seem suggestive based on more recent, scientifically accurate books [17] [18] [19]. The similarity between not only spiral rainbands of hurricanes and the spiral arms of a galaxy such as our own Milky Way, but also between both and other structures (albeit not with respect to 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 cyclones, 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 ${r}_{\text{eye}}\le r\le {r}_{max}$ if the cyclone has an eye, and perhaps somewhat beyond ${r}_{max}$). 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. That is, gravity must be construed as generating a negative pressure or tension.

If gravity generates tension, then space, or more correctly spacetime, must be capable of supporting tension. If space, or more correctly spacetime, is construed as a medium rather than as mere nothingness, then perhaps this tension could be construed as warping or curving space, or more correctly spacetime. Perhaps this might provide a physical interpretation for the statement: “Spacetime tells matter how to move, matter tells spacetime how to curve [21].” 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 medium), and since electromagnetic radiation is comprised of transverse waves, can space, or more correctly spacetime, 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] ? Concerning the latter point, the conventional viewpoint is, of course, that electromagnetic waves serve as their own medium, via the continual handoff of energy from transverse electric field to transverse magnetic field to transverse electric field... [27].

For an isolated spherically-symmetrical gravitator of radius
${r}^{\ast}$ and mass M for which Newtonian theory is sufficiently accurate for calculation of orbital velocity
${v}_{\text{orbit}}\left(r\right)$, escape velocity
${v}_{\text{escape}}\left(r\right)$, and gravitational potential
$\Phi \left(r\right)$ and hence of magnitude
$\left|\Phi \left(r\right)\right|$,^{FTNT16} i.e., for which General Relativity need not be employed for these purposes, applying Equations (4) and (5), at
$r\ge {r}^{\ast}$

${v}_{\text{escape}}\left(r\right)={\left|2\Phi \left(r\right)\right|}^{1/2}={\left(\frac{2\mathbb{G}M}{r}\right)}^{1/2}\Rightarrow n=-\frac{1}{2}\Rightarrow G\propto {r}^{-2}\mathrm{.}$ (39)

(The universal gravitational constant $\mathbb{G}$ should not be confused with the pressure gradient G.)

At
$r\ge {r}^{\ast}$ for an isolated spherically-symmetrical gravitator
${v}_{\text{escape}}\left(r\right)={2}^{1/2}{v}_{\text{orbit}}\left(r\right)$, so either
${v}_{\text{escape}}\left(r\right)$ or
${v}_{\text{orbit}}\left(r\right)$ can be used.^{FTNT16} In Equations (39)-(43) and the associated discussions we choose
${v}_{\text{escape}}\left(r\right)$ for closer analogy with the maximum speed attainable consistent with conservation of energy^{FTNT16} as per Bernoulli’s equation for fluid flow [6] [7] :^{FTNT2} recall Equations (6)-(9) and the associated discussions. But we note that, applying Equation (2) and that the gravitational force on a test particle of mass
$m\ll M$ is
${F}_{\text{grav}}=\mathbb{G}Mm/{r}^{2}$,
${v}_{\text{orbit}}\left(r\right)$ for circular orbits can be derived similarly to Equation (3):

${F}_{\text{centr}}={F}_{\text{grav}}$

$\Rightarrow \frac{m{\left[{v}_{\text{orbit}}\left(r\right)\right]}^{2}}{r}=\frac{\mathbb{G}Mm}{{r}^{2}}$

$\Rightarrow {v}_{\text{orbit}}\left(r\right)={\left(\frac{\mathbb{G}M}{r}\right)}^{1/2}={\left|\Phi \left(r\right)\right|}^{1/2}.$ (40)

Strictly, ${F}_{\text{grav}}$ 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.

Space and time intervals
$\mathbb{I}$ of unity (
$\mathbb{I}=1$) in gravity-free space (for the case of an isolated spherically-symmetrical gravitator in the limit
$\frac{r}{{r}^{\ast}}\to \infty $) are diminished in weak-gravitational-field limit (
${v}_{\text{escape}}\ll c\iff \left|\Phi \right|\ll {c}^{2}$) to^{FTNT17}

$\mathbb{I}=1-\frac{{v}_{\text{escape}}^{2}}{2{c}^{2}}=1-\frac{\left|\Phi \right|}{{c}^{2}}.$ (41)

For weak spherically-symmetrical gravitational fields (at $r\ge {r}^{\ast}$)

$\Phi \left(r\right)=-\frac{\mathbb{G}M}{r}\Rightarrow \left|\Phi \left(r\right)\right|=\left|\frac{\mathbb{G}M}{r}\right|$

$\Rightarrow \mathbb{I}\left(r\right)=1-\frac{\mathbb{G}M}{r{c}^{2}}.$ (42)

Qualitatively, we should expect a tension or negative pressure to “squeeze” or “pinch” spacetime, and hence to diminish both spatial and temporal intervals. Qualitatively, this is what obtains.^{FTNT17} Quantitatively, we may be on less certain ground if we try to relate
$\Delta P\left(r\right)$ corresponding to the degree of “squeezing” or “pinching” of spacetime,^{FTNT17} but let us try anyway. Can we draw the following analogy at
$r\ge {r}^{\ast}$, as per Equations (6) and (8), with the help of Equations (39) and (42)?

${v}_{\text{escape}}\left(r\right)={\left(\frac{2\mathbb{G}M}{r}\right)}^{1/2}\stackrel{?}{\iff}{v}_{\text{escape}}\left(r\right)={\left[\frac{2\Delta P\left(r\right)}{\rho \left(r\right)}\right]}^{1/2}$

$\Rightarrow {\left(\frac{2\mathbb{G}M}{r}\right)}^{1/2}={\left[\frac{2\Delta P\left(r\right)}{\rho \left(r\right)}\right]}^{1/2}\mathrm{?}$

$\Rightarrow \Delta P\left(r\right)=\frac{\mathbb{G}M}{r}\rho \left(r\right)\left(\text{spherical symmetry,}{v}_{\text{escape}}\ll c\iff \left|\Phi \right|\ll {c}^{2}\right)\mathrm{?}$

$\Rightarrow \Delta P\left(r\right)=\left|\Phi \left(r\right)\right|\rho \left(r\right)\left(\text{more general geometry,}{v}_{\text{escape}}\ll c\iff \left|\Phi \right|\ll {c}^{2}\right)\mathrm{?}$ (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 ${v}_{\text{escape}}\left(r\right)$ as per Equation (39) or ${v}_{\text{orbit}}\left(r\right)$ as per

Equation (40)? If ${v}_{\text{orbit}}\left(r\right)$ as per Equation (40) had been employed, then

${\left(\frac{2\mathbb{G}M}{r}\right)}^{1/2}\to {\left(\frac{\mathbb{G}M}{r}\right)}^{1/2}$ in the first two lines of Equation (43) and there would

be an extra factor of $\frac{1}{2}$ immediately following the equal (=) sign in each of the

last two lines thereof. (b) In strong gravitational fields wherein
${v}_{\text{escape}}\left(r\right)$ and even
${v}_{\text{orbit}}\left(r\right)$ is a significant fraction of c and
$\left|\Phi \right|$ is a significant fraction of
${c}^{2}$, the Newtonian approximations as per Equations (39)-(43) and the associated discussions must be modified.^{FTNT18}

Yet, at least prima facie, our result of Equation (43) seems to be qualitatively reasonable. For a given $\left|\Phi \left(r\right)\right|$, the less massive the gravitator, the larger $\rho \left(r\right)$ must be for a given $\Delta P\left(r\right)$. At least prima facie, it seems qualitatively reasonable that, for a given $\left|\Phi \left(r\right)\right|$, a more spatially compact and denser gravitator―a larger $\rho \left(r\right)$ ―should correspond to a larger $\Delta P\left(r\right)$.

For the region of 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 gravitator, because the mass of the Milky Way 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.^{FTNT19-21} In the region of Milky Way in the vicinity of the Sun’s orbit,
${v}_{\text{orbit}}~{r}^{0}$, i.e.,
${v}_{\text{orbit}}$ ~ independent of r.^{FTNT19-21} [By Equations (39) and (40) and the associated discussions this implies that in the vicinity of the Sun’s orbit

${M}_{\le r}/r=\left(4\text{\pi}\langle {\rho}_{\le r}\rangle {r}^{3}/3\right)\xf7r=4\text{\pi}\langle {\rho}_{\le r}\rangle {r}^{2}/3\approx \langle {\rho}_{\le r}\rangle {r}^{2}\sim $ independent of r (the subscript $\le r$ means “within r”).] Hence applying Equations (4) and (5):

${v}_{\text{orbit}}~{r}^{0}\left({v}_{\text{orbit}}~\text{independent of}r\right)\Rightarrow n\approx 0\Rightarrow G\text{\hspace{0.17em}}\text{approximately}\propto {r}^{-1}\mathrm{.}$ (44)

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 [28].^{FTNT22}

5.2. Cylindrically-Symmetrical Gravity

For comparison, let us briefly consider the gravitational field of a long cylindrical mass M of radius ${r}^{\ast}$ and length $l\gg {r}^{\ast}$, at radial distance r from its central axis and at the center of its length ( $l/2$ from both ends), with ${r}^{\ast}\le r\ll l/2$.

The gravitational force on a test particle of mass
$m\ll M$ in this cylindrical case can be derived from that in the more usual spherical case (a simple example of Gauss’ Law^{FTNT23}):

${F}_{\text{grav,sph}}=\frac{\mathbb{G}Mm}{{r}^{2}}=\frac{4\text{\pi}\mathbb{G}Mm}{4\text{\pi}{r}^{2}}=\frac{4\text{\pi}\mathbb{G}Mm}{\mathbb{S}}\left(\text{sphericalcase}\right)$

$\Rightarrow {F}_{\text{grav,general}}=\frac{4\text{\pi}\mathbb{G}Mm}{\mathbb{S}}\left(\text{generalcase}\right)$

$\Rightarrow {F}_{\text{grav,cyl}}=\frac{4\text{\pi}\mathbb{G}Mm}{2\text{\pi}rl}=\frac{2\mathbb{G}Mm}{rl}=\frac{2\mathbb{G}{\rho}_{\text{linear,cyl}}m}{r}\left(\text{cylindricalcase}\right).$ (45)

In Equation (45) $\mathbb{S}$ is the area of a Gaussian surface everywhere normal or perpendicular to the lines of force [ $\mathbb{S}=4\text{\pi}{r}^{2}$ for spherical symmetry and $\mathbb{S}=2\text{\pi}rl$ for cylindrical symmetry (neglecting the ends of a cylinder)] and ${\rho}_{\text{linear,cyl}}=M/l$ is the linear mass density of a cylinder. Setting ${F}_{\text{grav,cyl}}={F}_{\text{centr}}$ for orbital motion in this cylindrical case, and applying Equation (2), yields [in similarity with Equations (3) and (40)]

${F}_{\text{centr}}=\frac{m{v}_{\text{orbit}}^{2}}{r}={F}_{\text{grav,cyl}}=\frac{2\mathbb{G}{\rho}_{\text{linear,cyl}}m}{r}$

$\Rightarrow {v}_{\text{orbit}}={\left(2\mathbb{G}{\rho}_{\text{linear,cyl}}\right)}^{1/2}.$ (46)

Thus in this cylindrical case ${v}_{\text{orbit}}$ is independent of r, i.e., ${v}_{\text{orbit}}\propto {r}^{0}$. Hence applying Equations (4) and (5)

${v}_{\text{orbit}}\propto {r}^{0}\left({v}_{\text{orbit}}\text{independentof}r\right)\Rightarrow n=0\Rightarrow G\propto {r}^{-1}\mathrm{.}$ (47)

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 $\Phi \left(r\right)$ for the cylindrically-symmetrical case is not as straight-forward as for the spherically-symmetrical case. In the region ${r}^{\ast}\le r\ll l/2$ :

$\Phi \left(r\right)=\Phi \left({r}^{\ast}\right)+2\mathbb{G}{\rho}_{\text{linear,cyl}}\mathrm{ln}\frac{r}{{r}^{\ast}}.$ (48)

For an infinitely long ( $l\to \infty $) cylindrical gravitator, for any finite ${\rho}_{\text{linear,cyl}}$, however small, $\Phi \left(r\right)-\Phi \left({r}^{\ast}\right)$ diverges (albeit only logarithmically) with increasing r. Therefore, unlike in the spherical case, we cannot set $\Phi \left(r\right)\to 0$ in the limit $r\to \infty $. Thus not even the most powerful rocket, indeed not even light, can escape from an infinitely long sewing thread! Thus if $l\to \infty $ there is no Newtonian (weak-field) limit for a cylindrically-symmetrical gravitational field, not even for that of a sewing thread. For finite l in the weak-field limit ${F}_{\text{grav,cyl}}$ gradually changes from that given by Equation (45) if ${r}^{\ast}\le r\ll l/2$ to that for a spherically-symmetrical gravitational field as per Equation (40) if $r\gg l$ ; and $\Phi \left(r\right)$ from that given by Equation (48) to that given as per Equations (39)-(43).

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 [28].^{FTNT22}

6. 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 of pressure, 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 hence with fixed v, and 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
${r}^{-1}$, 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
${r}^{-1}$, 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
${r}^{-1}$, generation of kinetic energy falls short of matching frictional loss so v decreases with decreasing r. (Refer to Sections 1 and 2 as necessary.)

Although our main concern in this paper is with cyclostrophic flow, a brief comparison with generation of kinetic energy in geostrophic flow (straight isobars),^{FTNT9,10} in friction-balanced flow, 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 [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 hence with fixed v, and would never be able to move down a hill of pressure. If in geostrophic flow
$G/sin\varphi $ is constant (independent of pressure), generation of kinetic energy matches frictional loss so v is also constant (independent of pressure). If in geostrophic flow
$G/sin\varphi $ 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
$G/sin\varphi $ 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 flow, 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 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 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 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 steady-state river or ground-water flows, which are friction-balanced 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 friction-balanced river flow $gsin\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)$ is constant, generation of kinetic energy matches frictional loss so v is also constant (independent of elevation). If in friction-balanced river flow $gsin\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)$ increases downstream, generation of kinetic energy exceeds frictional loss so v also increases downstream. If in friction-balanced river flow $gsin\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)$ decreases downstream, generation of kinetic energy falls short of matching frictional loss so v also decreases downstream. [As noted in the paragraph immediately following Equation (37), in most steady-state rivers on Earth $gsin\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta \xf7\left(\mathcal{P}/\mathcal{A}\right)$, and hence also v, increases slightly downstream.] If in friction-balanced ground-water flow $gsin\theta /{\mathcal{C}}^{\u2033}\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta /{\mathcal{C}}^{\u2033}$ is constant, generation of kinetic energy matches frictional loss so v is also constant (independent of elevation). If in friction-balanced ground-water flow $gsin\theta /{\mathcal{C}}^{\u2033}\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta /{\mathcal{C}}^{\u2033}$ increases downhill, generation of kinetic energy exceeds frictional loss so v also increases downhill. If in friction-balanced ground-water flow $gsin\theta /{\mathcal{C}}^{\u2033}\stackrel{\theta \ll 1\text{\hspace{0.05em}}\text{rad}}{\doteq}g\theta /{\mathcal{C}}^{\u2033}$ decreases downhill, generation of kinetic energy falls short of matching frictional loss so v also decreases downhill. Note the similarity of the relation between the variation of the ratio of the driving force to the frictional retarding force and the variation of v in the geostrophic case and the friction-balanced cases, despite the Coriolis force being of first importance in the geostrophic case but negligible or zero in the friction-balanced cases.

In cyclones with eyes―vortices as per our construed definition in the first paragraph of Section 1―(tornadoes, dust devils, waterspouts, and 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 [6] [7] ,^{FTNT2} as per Equations (6) and (8). 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 more, usually much more, than the minimum (
$\propto {r}^{-1}$ in cyclostrophic flow) required for v to increase in the direction of decreasing pressure―assuming at least some friction to allow the wind to move down hills, or rather down into pits, of pressure―in cyclones with eyes, but not in extratropical synoptic-scale weather systems. For example, unlike in cyclones with eyes in general and hurricanes in particular, in extratropical cyclones the pressure gradient often does not steepen towards lower pressure at all. Thus maximum winds in extratropical cyclones are usually considerably slower than those of hurricanes, and typically occur at intermediate distances between the center and the outer periphery, rather than near (but not at) the center (at the eye wall) as in cyclones with eyes in general and hurricanes in particular. At the very least most extratropical cyclones lack eyes; perhaps all do. All anticyclones lack eyes. (Of course, all cyclones are pits of pressure and all anticyclones are hills of pressure.)

In friction-balanced flow, most commonly, at least approximately, generation of kinetic energy matches frictional loss, so v remains at least approximately constant (in steady-state 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 [6] [7] ,^{FTNT2} as per Equations (6) and (8).

The first paragraph of this Section 6 applies in gravitational cases too. The pits in these cases are gravitational potential wells, but so are, ultimately, the pits of pressure represented by cyclones and the hills of pressure represented by anticyclones. 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 $G\propto {r}^{-2}$ (increasing inwards faster than $\propto {r}^{-1}$) generation of kinetic energy exceeds frictional loss so v increases as a satellite spirals inwards towards decreasing r. In contrast, by Section 5.2 in the case of cylindrically-symmetrical gravitation $G\propto {r}^{-1}$, and hence generation of kinetic energy matches frictional loss so v remains constant as a satellite spirals inwards towards decreasing r. In a geometry where $G\propto {r}^{n}$ with $n>-1$ (increasing inwards slower than $\propto {r}^{-1}$) generation of kinetic energy falls short of matching frictional loss so v decreases as a satellite spirals inwards towards decreasing r.

7. Brief Concluding Remarks

In Section 2 we discussed 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 were briefly discussed. We considered only small atmospheric vortices, namely tornadoes, and dust devils, and small hurricanes at low latitudes, so that we could assume that the Coriolis force can be neglected, and 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 $\rho $. 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 $\rho $ is an excellent approximation. Brief comparisons with geostrophic and friction-balanced flows were provided in Section 4, again neglecting variations in fluid density $\rho $. In Section 5 we briefly 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 briefly 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 explained the variations of pressure gradients and gravitational gradients corresponding to generation of kinetic energy exceeding, equaling, and falling short of frictional dissipation. Expanding on the second paragraph of Section 3.1 where windmills are briefly discussed, the Appendix describes a simple method for maximizing power extraction from environmental fluid (air or water) flows. It also briefly explains why this method is in general not applicable to artificial (e.g., internal combustion) engines. Our overview of features and energetics of Earth’s environmental fluid flows, even though mostly 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.

Acknowledgements

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

Footnotes

Footnote 1: See Ref. [1] , Exercise 8.6 on p. 357.

Footnote 1A: There is an additional restriction on wind speed in anticyclones, which requires calms at their centers, 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 does not apply.

Footnote 2: See Ref. [2] , pp. 674-675.

Footnote 4: See: “Enhanced F Scale for Tornado Damage” (most recently revised in 2007) at https://www.spc.noaa.gov/faq/tornado/ef-scale.html.

Footnote 5: See: “Saffir-Simpson Hurricane Scale” (no date available) at https://www.nhc.noaa.gov/aboutsshws.php.

Footnote 6: See Ref. [3] , p. 152.

Footnote 7: See Ref. [4] , pp. 137-138.

Footnote 8: See Ref. [6] , Chaps. 18 and 20.

Footnote 9: See Ref. [1] , Section 7.2, especially Subsections 7.2.4-7.2.5.

Footnote 10: See Ref. [2] , Sections 2.4.1 and 3.2.2.

Footnote 11: See Ref. [1] , Section 8.2.3.

Footnote 12: See Ref. [11] , pp. 310-312.

Footnote 14: See also Ref. [6] , Section 26-4.

Footnote 15: See Ref. [4] , p. 4.

Footnote 16: See Ref. [6] , Chap. 13.

Footnote 17: See Ref. [25] , Sections 1.16, 9.1-9.3, 9.7, and Chaps. 8-11 (in Chap. 11 see especially Sections 11.2-11.6 and 11.9).

Footnote 18: See Ref. [25] , Chaps. 11-12, especially Section 12.2.

Footnote 19: See Ref. [17] , Section 14.1.6.

Footnote 20: See Ref. [18] , Section 24.3 (especially pp. 914-919), and pp. 951-956.

Footnote 21: See Ref. [19] , Section 10.3 (especially pp. 406-411), and pp. 443-448.

Footnote 22: See Ref. [12] , pp. 426-428 and 1206.

Footnote 23: See Ref. [6] , Chap. 23.

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 (air or water) flows; e.g., power extraction from the wind by a windmill, from the flow of a river by a waterwheel, etc. If for example a windmill or waterwheel is turning freely with no load imposed on it, so that it is not required to supply any torque $\mathcal{T}$ or power $\mathbb{P}$, it will spin at its maximum possible angular velocity ${\omega}_{max}$ in a given environmental fluid flow. (For an undershot waterwheel of radius $\mathcal{R}$ in a river flowing at linear velocity v, ${\omega}_{max}=v/\mathcal{R}$.) As the load imposed on a windmill or waterwheel is increased and it is required to supply increasing torque $\mathcal{T}$, its angular velocity $\omega $ will decrease monotonically. At maximum possible load, with it being required to supply maximum possible torque ${\mathcal{T}}_{max}$, its angular velocity $\omega $ will have decreased to zero, so again it will supply zero power. Thus the power

$\mathbb{P}=\mathcal{T}\omega $ (A1)

extracted from an environmental fluid flow will be maximized at intermediate values of $\mathcal{T}$ and $\omega $.

Let $\mathcal{T}$ be plotted as a function of $\omega $ on a graph whose origin is ( $\mathcal{T}=0$, $\omega =0$), with $\mathcal{T}$ increasing linearly upwards on the vertical axis and $\omega $ increasing linearly to the right on the horizontal axis. Now to maximize $\mathbb{P}$ :

$\text{d}\mathbb{P}=\text{d}\left(\mathcal{T}\omega \right)=\mathcal{T}\text{d}\omega +\omega \text{d}\mathcal{T}=0$

$\Rightarrow \mathcal{T}\text{d}\omega =-\omega \text{d}\mathcal{T}$

$\Rightarrow \frac{\text{d}\mathcal{T}}{\text{d}\omega}=-\frac{\mathcal{T}}{\omega}\text{\hspace{0.17em}}\text{attheoptimumpoint}\left(\mathcal{T}={\mathcal{T}}_{\text{opt}},\omega ={\omega}_{\text{opt}}\right)$

$\Rightarrow {\mathbb{P}}_{max}={\mathcal{T}}_{\text{opt}}{\omega}_{\text{opt}}\mathrm{.}$ (A2)

Thus $\mathbb{P}$ is maximized at the point on the $\mathcal{T}$ versus $\omega $ plot where the

positive slope $\frac{\mathcal{T}}{\omega}$ from the origin ( $\mathcal{T}=0,\omega =0$) to that point is equal in magnitude to the negative slope $\frac{\text{d}\mathcal{T}}{\text{d}\omega}$ of the tangent at that point. Hence this is

the optimum point ( $\mathcal{T}={\mathcal{T}}_{\text{opt}},\omega ={\omega}_{\text{opt}}$) on the $\mathcal{T}$ versus $\omega $ plot, corresponding to maximum power ${\mathbb{P}}_{\mathrm{max}}={\mathcal{T}}_{\text{opt}}{\omega}_{\text{opt}}$ extracted from an environmental fluid flow.

Thus a waterwheel or a windmill will achieve its maximum possible efficiency and maximum possible power output
${\mathbb{P}}_{\mathrm{max}}={\mathcal{T}}_{\text{opt}}{\omega}_{\text{opt}}$ if operating at this optimum point (
$\mathcal{T}={\mathcal{T}}_{\text{opt}},\omega ={\omega}_{\text{opt}}$) on its
$\mathcal{T}$ versus
$\omega $ plot, in accordance with Equation (A2). For a windmill the upper bound on this maximum possible efficiency is generally construed to be the Betz limit [8] [9] [10].^{FTNT3} (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 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 wind turbines [9] [10]. An equivalent of the Betz limit for horizontal 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 efficiency and maximum possible power output
${\mathbb{P}}_{\mathrm{max}}={\mathcal{T}}_{\text{opt}}{\omega}_{\text{opt}}$ if operating at this optimum point (
$\mathcal{T}={\mathcal{T}}_{\text{opt}},\omega ={\omega}_{\text{opt}}$) on its
$\mathcal{T}$ versus
$\omega $ plot, in accordance with Equation (A2).

It should be noted that novel systems to extract energy from the wind are being developed. These include flying windmills [29] and wind-harvesting systems with no moving parts [30] [31] [32].^{FTN24} The latter share with vertical-axis wind turbines balanced weight distributions about their centers and also occupying less space than horizontal wind turbines―in addition to having no moving parts at all rather than merely fewer moving parts than horizontal wind turbines. Perhaps Equations (A1) and (A2) could apply for nonrotary [30] [31] [32] wind-energy systems^{FTNT24} if appropriate analogs of
$\mathcal{T}$ and
$\omega $ were employed, and perhaps an equivalent of the Betz limit might (or might not) exist for such systems.

In extraction of power by rotary devices (e.g., waterwheels, conventional horizontal-axis windmills, vertical-axis windmills, and flying windmills [29] ) from environmental fluid flows, $\mathcal{T}$ always decreases monotonically with increasing $\omega $, because environmental fluid flows are independent of ( $\mathcal{T}\mathrm{,}\omega $) of the power-extracting device. Thus power extracted from environmental fluid flows is always maximized in accordance with Equation (A2). By contrast, for artificial engines (e.g., internal combustion engines) this need not be the case, because fuel flow to an artificial engine can in general vary as a function of ( $\mathcal{T}\mathrm{,}\omega $) of an engine [33]. Thus while Equation (A1) is always valid for any rotary system, Equation (A2) is in general not valid for maximization of artificial-engine power output [33]. Indeed Equation (A2) might not be realized at any point on the $\mathcal{T}$ versus $\omega $ plot for an artificial engine [33].

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Open Journal of Fluid Dynamics*,

**8**, 216-247. doi: 10.4236/ojfd.2018.82015.

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