# Applying the Action Principle of Classical Mechanics to the Thermodynamics of the Troposphere

^{1}

^{2}

^{*}

## Abstract

**:**

_{t}≡ ∫mvds, J-sec) is used to show how equilibrium temperatures are achieved from statistical equality of mechanical torques (mv

^{2}or mr

^{2}ω

^{2}); these are exerted by Gibbs field quanta for each kind of gas phase molecule as rates of translational action (d@

_{t}/dt ≡ ∫mr

^{2}ωdϕ/dt ≡ mv

^{2}). These torques result from the impulsive density of resonant quantum or Gibbs fields with molecules, configuring the trajectories of gas molecules while balancing molecular pressure against the density of field energy (J/m

^{3}). Gibbs energy fields contain no resonant quanta at zero Kelvin, with this chemical potential diminishing in magnitude as the translational action of vapor molecules and quantum field energy content increases with temperature. These cases distinguish symmetrically between causal fields of impulsive quanta (Σhν) that energize the action of matter and the resultant kinetic torques of molecular mechanics (mv

^{2}). The quanta of these different fields display mean wavelengths from 10

^{−4}m to 10

^{12}m, with radial mechanical advantages many orders of magnitude greater than the corresponding translational actions, though with mean quantum frequencies (v) similar to those of radial Brownian movement for independent particles (ω). Widespread neglect of the Gibbs field energy component of natural systems may be preventing advances in tropospheric mechanics. A better understanding of these vortical Gibbs energy fields as thermodynamically reversible reservoirs for heat can help optimize work processes on Earth, delaying the achievement of maximum entropy production from short-wave solar radiation being converted to outgoing long-wave radiation to space. This understanding may improve strategies for management of global changes in climate.

## 1. Introduction

^{2}ωdϕ ≡ mr

^{2}ωδϕ, J.s, with rdϕ or ds spatial motion). In this form, action has the same dimensions as the vector state of angular momentum (mr

^{2}ω) since the differential for the phase angle (dϕ ≡ ωdt) is dimensionless, signaling a dynamic change in spatial configuration. The action (h) of quanta at all wavelengths has the same value, a fundamental natural constant defined [3] by Planck (6.62607 × 10

^{−34}J.s). Action of material particles is quantified as a function of radial curvature proportional to 1/r using radial coordinates; indeed, sustained rather than brief rectilinear motion is regarded as very unusual, except for the trajectories of quanta.

## 2. Basic Theory and Methods

#### 2.1. Complementary Fields for Virtual Quanta and Dynamics of Material Particles

- We use data from our previous action revision of the Carnot cycle [2] to examine the thermodynamics of the working fluid in expressing power from heat engines in terms of variable quantum fields. Using Carnot’s cycle, the study showed how the difference between the quantity of calorique obtained from a hot source and that absorbed by a colder sink gives the maximum power output possible from any heat engine while returning the engine to its initial thermodynamic state. The total energy (U) of the working fluid is not the source of the external work performed, but merely a scaffold for the variation in Gibbs quantum fields between the 2 distinct isothermal states that allow maximum work. A method to define the mean wavelength and frequency of the field quanta is given.
- By applying action mechanics to sustain the thermal and gravitational structure of the troposphere [7], we show how the virial theorem establishes the temperature gradient with altitude rather than adiabatic expansion. The quantum field characteristics of increasing altitude are calculated in this article, partitioned between translational and rotational energies.
- We illustrate the role of the vortical action (ΣmR
^{2}ω) of concerted molecular flow in anticyclones and cyclones as a higher degree of freedom storing thermal energy as vortical work, capable of warming the surface by turbulent friction. - We estimate the maximum potential of vortical wind power in the Earth’s atmosphere, already foreshadowed in an article now in review explaining a new method to calculate the maximum power of wind turbines [8].
- We explain the destructive power of tropical cyclones derived from solar heat consumed in the evaporation of seawater, which also uses vortical energy dependent on quantum impulses at the molecular level.

#### 2.2. The Principle of Least Action Exemplified in Gibbs Field Mechanics

^{2}ω), stressing that it is a scalar function indicating change in configuration. It includes a spatial variation in the dimensionless property of angular motion (@ ≡ mr

^{2}ωδϕ ≡ Jδϕ) by reference to a local couple and distant stars. Symmetry factors may also be needed to account for symmetrical pairing of particles or repeated structures. Any variation in action as the rate of change of this angle (dϕ/dt = ω) can be expressed as a variation in torque (mr

^{2}ω

^{2}).

_{1}and r

_{2}are radii to the Earth’s gravitational center. Analogous processes are assumed to occur in many-bodied systems of molecules, with quantum fields generating the same torques and temperatures for each molecular species.

^{2}) decreases with height or radius, the Lagrangian (and its integral action) will include the same torque-like forms as in Equation (3), just as in Equation (1). This theoretical account is claimed as equally valid for many-bodied molecular systems in a microcanonical heat bath variable in temperature and torque, as for a gravitational system.

#### 2.3. Mathematical Basis and Procedure for Estimating Dynamic Action and Quantum Fields

_{2}, the complex macroscopic variables of temperature and volume or density expressing entropy (S) and heat (ST) of statistical mechanics have been integrated in simplified form [1] as functions of translational action (@

_{t}= mrv) and rotational action (@

_{r}= Iω), including factors for symmetry (z

_{t}, σ). z

_{t}includes a term for the ratio of mean to root mean square velocity. This process involved a reinterpretation [1] of the partition functions for translation and rotation, given in most textbooks on statistical mechanics.

_{e}of action ratios (@/ħ). Planck’s reduced constant (ħ = h/2π) is used in the denominator to measure the mean levels of the translational or rotational quantum states. Here, e indicates a mathematical exponent for internal energy and the subscripts refer to translation (t), rotation (r), and the 3 moments of rotational inertia (A, B, C) for 3-D molecules. This is classical statistical physics mechanics recast in a more holistic form. The heat capacity exponential term depends on whether the system volume is constant (C

_{v}) or is able to expand at atmospheric pressure (C

_{p}). In the following analysis, molecular pressure (p) for gases is expressed as a mean value for each species by the ideal gas law, shown in Equation (6), where N is the number of molecules per unit volume (1/a

^{3}).

^{3}= NkT

^{2}/3, where a is the mean cubic separation of molecules (r = a/2); each molecule is regarded as confined to its mean specific volume of a

^{3}.

_{v}T, J], or enthalpy [H, C

_{p}T, J] and Gibbs energy [-G, J], though using lower case as energy per molecule. It is easier to consider average values for molecules. In a variable pressure volume system undertaking reversible mechanical work such as the Carnot cycle, if estimating the molecular energy (e), c

_{p}T must be replaced by c

_{v}T with c

_{v}equal to 3/2 for a monatomic gas such as argon. No work is undertaken against the atmosphere, with the internal field energy varying with internal pressure reversibly, with respect to the back pressure from the external work. For maximum efficiency, the whole cycle must be performed reversibly, as Carnot defined.

_{t}T = c

_{v}T + kTln[(@

_{t}/ћ)

^{3}] = c

_{v}T + 3kTln[(n

_{t})] = e − g

_{t}

_{t}) is 0 at absolute 0, where the temperature is 0 K and becomes progressively more negative as the temperature increases, and the Gibbs field spontaneously gains quantum energy. This is consistent with the Maxwell relationship where ΔG equals ΔH − TΔS and spontaneous reactions have negative changes in Gibbs energy, as is the case when entropy increases. In Equation (7), kT is equal to mv

^{2}/3 for translational kinetic energy.

_{i}/c) and the reversible momentum exchange for the material particles (Σmv

_{i}) is equal within the same Brownian [5] or random walk matrix. Per se, neither material particles nor quanta exchange momentum directly and particles only do so in collisions by way of the far swifter intervening quanta. In terms of the rate of action impulses, establishing the number of quanta keeping the material particles separated is effectively magnified by the factor c/v

_{i}; so, a simple comparison of the ratio of momentum, either (hv

_{i}/cmv

_{i}) or (h/λ

_{i}mv

_{i}), will be greater by this factor than is needed for quanta in the field. This hypothesis will be tested during the various applied exercises as follows.

- The mean translational (or rotational) action (@
_{t}= mrv_{t}) of molecules is estimated for the molecular field based on macroscopic concentration and temperature, including any effect of symmetry that multiplies the probability of field energy interacting with particular groups, reducing free paths. The more symmetry exhibited in a mechanical system, the lower the action and the field energy needed to sustain the system [1]. To estimate translational action (@_{t}= mrv_{t}), the mean velocity is required rather than the root mean square velocity, which is approximately 1.09 times less. The simple methods used in action mechanics to calculate entropy and absolute Gibbs energy based on molecular properties were applied in a paper [6] examining thermodynamics of H_{2}and its lysis to hydrogen atoms at the temperature of the sun’s surface. Its difficult formation by thermal decomposition of water above 4500 K and by a much easier reversible formation from ammonia in the Haber process near 400 K were shown. - The mean number (n
_{t}) for translational quantum microstates per molecule for current mechanics is extracted by the ratio of the mean molecular action to Planck’s reduced quantum of action (n = @/ћ). For all cases examined in this article, this ratio exceeds unity by a significant margin, indicating a high entropy for this degree of freedom. For this reason, these translational processes all behave classically given the low rate of occupancy of quantum microstates. - The absolute value of the translational Gibbs energy (g
_{t}) is then estimated as a logarithmic function of the number of quantum microstates, as published previously [1,2,6]. As this value becomes more negative, the field quantum energy increases, as anticipated by the second law of thermodynamics. - The mean value of virtual quanta in the field is then calculated (hv = −g
_{t}/n_{t}), enabling the virtual frequency and wavelength in the field to be estimated. Peak values for translational quanta will reflect the vis viva, twice the kinetic energy for the Carnot cycle (mv^{2}= 3kT). For other processes, such as the dynamics of air molecules in wind, the vis viva involved is very low and according to wind speed, indicating a very low temperature using the 1-D relationship mv^{2}= kT.

## 3. Results and Discussion

#### 3.1. Revising the Carnot Cycle as a Basis for a Gibbs Action Field

_{v}T) plus the absolute energy of the field (−g

_{t}), which becomes more positive as its quantum state increases. Removing temperature (T) in Equation (7) gives the absolute entropy per molecule of the quantum state under the current environmental conditions of temperature and pressure. Table 1 reproduces and extends data from our earlier article [2]; shown are relationships between matter and quantum fields relevant to all four stages proposed by Carnot as reversible, determining the most efficient generation of power in the heat engine cycle. The table shows that Carnot’s formal explanation of the cycle using caloric is consistent with quantum theory, with its modern surrogate shown as mean negative Gibbs energy per molecule (−g

_{t}). Carnot specifically indicated that the maximum possible work was equal to the second differences of a–a’ or b’–b, where a, b’, a’, and b were primary differences between absolute Gibbs energy values calculated for argon and nitrogen shown in Table 1. For two working materials as ideal gases, the following conclusions from the heat engine cycle are made, considering the impulsive quantum properties of the working fluid as causal. Most formulae, such as the Schrödinger wave equation, estimate quanta absorbed or emitted as the difference between states, but Table 1 gives their absolute mean values. Note that these values for translational action (@

_{t}) are corrected here for a simple programming error in reference [2] that underestimated action by a factor of 1.47.

- At all four stages of the cycle, the relative action (@) of the working fluid calculated indicates its entropy state according to Equation (7). Gibbs energy (G
_{t}or g_{t}) is always zero or negative, decreasing from minimum action near absolute zero K. Uniquely, action mechanics quantifies the Gibbs field here as mean numbers of virtual quanta needed per molecule to sustain their temperature and pressure. - Atmospheric pressure is not relevant to the enclosed Carnot cycle, so from Equation (7) all effects of changes in pressure in the cycle can be calculated as changes in Gibbs energy calculated from macroscopic temperature and pressure, given these are equivalent [2,6]. Shown in Table 1, the field of virtual quanta (Σhv) contains almost 10 times as much field energy (largely provided in melting and vaporization) as the kinetic energy of the material particles, sustaining molecular torques (mv
^{2}) and material pressures. - Each turn of the Carnot cycle shown in Table 1 is assumed to absorb kT of heat from the hot source of quanta appropriate for the temperature and pressure and the same quantity kT removed at the colder sink as different quanta of lower frequency.
- The pressure values shown in the table also produce the ratio of torque intensity per unit volume (mv
^{2}/3a^{3}or kT/a^{3}) to the negative Gibbs energy density (−g_{t}/a^{3}) or mean density of virtual quanta held within the mean volume a^{3}occupied by each molecule. For argon, this energy ratio is constant for transitions in adiabatic or isentropic states with no change in heat content. Where isothermal processes at constant temperature (or torque) occur, there is a change in this ratio as heat is added or removed. - For nitrogen, the interaction between quantum cells for translation and rotation requires that the product of the quantum densities, shown as (n
_{t}^{3}× j_{r}^{2}), respectively, in the table, must remain constant for adiabatic processes that are isentropic, exhibiting constant action.

- The ratio for wavelength of virtual quanta and the material radial motion (λ/2πr; r = a/2) of approximately 10
^{5}for the gases is indicative of the ratio between the speed of light (λν = rω) and speed of the Brownian spiral of gas molecules. This can be visualized as the frequency of the conjugate quanta being of a similar order to that of the orbital frequency of the molecules but with the photon’s impulse cycling on a much longer radius, proportional to the ratio of speeds (c/v). - Table 1 also illustrates the correspondence for both argon (mass 40) and nitrogen (mass 28) of the ratio of the cumulative quantum impulse (nh/λ) and the dynamic impulse (Σmv = Σmrω) per molecule. This is a factor near 1 × 10
^{−5}, the inverse of the ratio of the speed of light to that of the molecules. In calculating translational action of molecules [2] it is necessary to make two corrections. One, a factor of 1/1.09 corrects root mean square velocity from the Maxwell distribution (3kT = mv^{2}) to mean velocity. The second corrects action for symmetry to avoid double counting of molecules (1/2). For cubic translation, this is an overall factor of 1/10.2297. This correction then allows the entropy calculated to match that for third law experiments in the literature. This correction factor (z_{t}) was initially established empirically [1], then interpreted rationally [2]. Overall, it allows the density of quanta needed to sustain the system to fall by a factor of 2.3205. - Another possible source for lack of correspondence in matching action impulses between quanta and molecules is that phase space for position and momentum can never exactly match true action space. The ideal coordinate system may not be Cartesian phase space since this separates variables (mv and r) that must be combined when quantized. A radial or polar system (r, ϕ, θ) is needed [7], but one that recognizes that changes in position in 3-D is absolutely quantized as jumps in the space of objects from one locus to another. There is no such thing as a smooth curve in nature for translation of rotation except by perception within the space of views, as explained a century ago by Jean Nicod [10].

#### 3.2. Thermodynamic Stabilization with Altitude for Atmospheric Gases by Quantum Fields of Molecules in Air

^{2}ω as used for calculating translational entropy in heat engines, shown in Table 1, the following relations by equating thermodynamic and gravitational pressure with altitude can be derived. In Equation (8), a represents the mean length of the side of a cube occupied by each different molecule, and r represents the mean radial separation which is half of that value. Then M is equal to Nm, the total mass of n molecules in the atmospheric column above a square with base of side a cm, of weight Nmg, assuming the value of gravity (g) is invariant in the troposphere. It is assumed that the inertial force $mr{\omega}^{2}$ provides the internal pressure on the six faces of the cube of side a, tending to equilibrate with the gravitational pressure or weight (Mg) per unit area (a

^{2}) of the atmosphere.

^{2}exerted in each six-faced cell of side a by the translational motion of each molecule. The primary gravitational pressure from the weight of air is exerted once only downwards and not to all six cardinal points. The thermodynamic relationship is statistical according to the Maxwell–Boltzmann distribution, with the molecular velocities having values statistically varying around the root mean square velocity which is characteristic of the temperature. Incidentally, the hydrostatic or isobaric requirement that the pressure is a function of density is only true for an isothermal atmosphere. In a real atmosphere with a tropospheric temperature gradient with an altitude of a little more than 6.5 K per km [7], pressure also varies as a function of temperature.

_{n}is equal to nkδT, yielding a lapse rate of δT/h

_{n}or mg/nk rather than mg/C

_{p}, where the divisor is the heat capacity of air at constant pressure. This new formula gives a lapse rate of temperature with an altitude of 6.9 C per km, close to the observed value and is only slightly reduced by heat released on condensation of water vapor. In the formula, m is the mean molecular weight of the gases in Daltons (28.97 for air) number-density weighted, where n indicates the degrees of freedom of action or kinetic motion able to contain heat, usually each k/2, although quantum effects of vibration can modify this freedom affecting n. For monatomic molecules such as argon, n is three. For diatomic molecules such as nitrogen and oxygen in their ground states it is five, and for simple polyatomic molecules found in the atmosphere it is six at ambient temperatures. However, the actual temperature gradient with altitude will be a cumulative variable determined by the complex properties of other gases in the atmosphere, their mixing ratios, and other local environmental factors such as temperature that may affect the vibrational heat capacity or quantum state. This is not a significant issue for the major diatomic gases in the atmosphere of Earth but would be on Venus [7] with its surface temperature in the vicinity of 740 K, with carbon dioxide as the major gas.

^{3}= 8Nr

^{3}) or density and temperature of molecules as action (mrvδϕ), illustrated in Section 3.1. In Table 2, estimates are given for thermodynamic properties of atmospheric N

_{2}as translational (n

_{t}) and rotational (j

_{r}) molecular action. Activation of vibrational states for N

_{2}in the atmosphere are negligible, as shown previously [1], given the high frequency and energies involved. If required, vibrational Gibbs energy can be estimated from the statistical component of vibrational entropy together with the zero-point energy Nhν/2 of 14.115 kJ per mol. More than 75% of the energy content indicated for N

_{2}is required to sustain its translational Gibbs field, shown in Table 2, with resonant quanta in the range 3.8 to 2.2 × 10

^{−22}J of frequency 5.7 × 10

^{11}to 3.32 × 10

^{11}Hz and wavelengths from 523 to 904 μm. N

_{2}contributes almost nothing [1] to the thermal emission to space in the infrared and far infrared up to 100 μm wavelengths, unlike water and other greenhouse gases.

_{2}molecules, Table 2 shows estimates with altitude for entropy per mole (S), absolute Gibbs quantum state levels (n

_{t}), the total heat energy required to reach the state (ST), and the mean value of virtual field quanta (hv). Vibrational energy for N

_{2}is negligible. To estimate peak wavelengths of quanta in the field energy, the virtual quantum value (hv) is divided by h to obtain frequency (v) then inverted and multiplied by the speed of light c to obtain the wavelength. Thus, at the surface temperature and pressure, it can be shown that 1.14833 × 10

^{26}quanta per mol of peak translational energy 3.809 × 10

^{−22}J support the kinetic activity of N

_{2}with a frequency of 5.74848 × 10

^{11}(574.849 GHz) with a peak frequency at a wavelength of 521.52 μm. At 12 km of altitude, the peak wavelength is almost twice as long at 901.704 μm, though there are more quanta per molecule. The table also shows absolute values for the Gibbs field energy per mol, a property that is always negative and becomes more so as spontaneous processes occur that increase the entropy.

#### 3.3. Vortical Action as High-Level Atmospheric Thermodynamics in Anticyclones

_{t}) of air molecules in anticyclones concerted in motion as wind velocity is estimated as mrv from knowledge of the mass of a material particle, the radial separation r equal to d/2, where d is the diameter of the anticyclone (Figure 1 and Figure 2). The number of quantum levels is estimated using division by the reduced Planck’s constant of action (h/2π = 1.054 × 10

^{−34}) (n

_{t}= mrv/ћ), with a symmetry factor of two for symmetrical partners. The Gibbs vortical energy per matter cell a

^{3}is then estimated using the logarithm of the quantum number multiplied by the appropriate torque factor. Frequency and wavelength are then easily determined. As with the Carnot cycle, the ratio of mean quantum wavelength and that of the radius to the center of the anticyclone is of the same magnitude as the ratio of the mean velocity of the molecule to the speed of light. The total vortical Gibbs energy is obtained from the product of number of molecules per cubic meter and the number of molecules.

^{2}of downwelling radiation is returned from the atmosphere, which explains the blackbody temperature of the Earth’s surface. It is proposed that instead of net radiation from higher in a colder atmosphere to the surface, vortical action in anticyclones is generated as work processes in air facilitated by Coriolis accelerations in each hemisphere. This work process requires significant absorption of heat radiated from the surface in greenhouse processes involving mainly water and carbon dioxide without rises in temperature. As shown in the Carnot cycle, any increase in freedom of relative translational motion of molecules increases the heat capacity of the gas phase. For anticyclones, this allows turbulent friction processes nearer the surface to release heat in the boundary layer of the lower atmosphere (Σhv) to the extent (h = a + e + c + g = Σhv) of approximately 330 watts per m

^{2}[2] as a global average rather than by radiation from a colder atmosphere to the surface, in accordance with the second law of thermodynamics. The decreasing wind speed near the surface regarded as vorticity represents the loss of power with wind speed, warming air, and causing spectral radiation proportional to temperature.

^{4}) from the surface and outgoing longwave radiation (o), with the atmosphere warmed by solar absorption by water in air (a), latent heat of evapotranspiration at the surface followed by condensation under convection (e), thermal conduction from the surface (c), and the greenhouse effect itself (r − o). The work of vortical action and energy (v) provides a mechanism for turbulent release of heat as radiation near the surface. No conflict with the second law of thermodynamics is required, solving the objection to net downwelling radiation from a colder source. As shown in Table 3, a wind speed of 10 m per sec contains a vortical energy of 1.47 × 10

^{3}J per m

^{3}of air in wind, many times greater than the kinetic energy, with an additional 2.4 MJ per m

^{3}of thermal energy required for air to be heated from 0 K to 298 K [1].

#### 3.4. Estimation of Power Produced by Wind Turbines from Vortical Energy in Anticyclones

^{6}m) from an anticyclone center (Figure 1 and Figure 2). The analysis considers the wind impacting the rotor blades as well as that passing through the circle of blade rotations. Only in the latter case does the kinetic energy available to the blades exceed the radial action estimate. By comparison, the maximum vortical power estimated is more than five times greater and is restricted to the blade area.

^{2}) at a 1000 km radius, estimates for vortical energy available are almost two orders of magnitude greater than that of kinetic energy.

^{−38}J with a wavelength of 2.6627 × 10

^{12}m. This wavelength is more than 10

^{6}greater than the corresponding material radius. This means that the curvature of the oscillating longitudinal motion is relatively linear and the action velocity for the molecules is approximately 10

^{6}less than the velocity of light. Differences in Gibbs energy per molecule with wind speed are easily calculated. Methods are suggested in our articles [2,6,8] to test the vortical energy field hypothesis using appropriate sensors, allowing release of this energy under turbulent conditions. If confirmed, this could be an important source of regional warming and land dehydration [8], possibly raising fire risk from wind farms. A 100 MW windfarm could raise the temperature of air downwind by the turbulent release of vortical energy up to 2 °C, increasing evapotranspiration for many kms. This prediction [8] is recommended to be tested as a matter of diligence regarding the location of windfarms.

#### 3.5. Power in Tropical Cyclones Estimated by Heat from Volatilization on the Ocean Surface and Convective Condensation of Water at the Eyewalls

^{2}ω) will be approximately constant across the cyclone, given the inverse square radial distribution of radiation from convective condensation at the eyewall, generating a radially acting Gibbs energy field as mv

^{2}ln[n

_{v}] as shown in Table 5. Furthermore, considering the cyclone as simultaneously rotational and convective, conservation of angular momentum will ensure constant Σmr

^{2}ω ensuring intensified velocity (rω) nearer the eyewall. The heat of vaporization at an estimated rate matches the vortical Gibbs energy and power generated, as shown in Table 5. Vortical entropic energy is directly provided by infrared radiation from condensing water in the convective eyewall of a tropical cyclone. The model shown in Figure 4 exhibits vortical action and energy, as shown in Table 5. Note the 70-fold ratio of energy density in the Gibbs field compared to that of kinetic pressure, similar to that observed for wind driving turbines to develop electrical power (Table 4).

## 4. General Discussion

^{2}/2), the static pressure energy P (Σmv

^{2}/3 = pV), and gravitational potential energy, regarded overall in steady flow as constant.

^{2}/2 + P + ρgh = K

^{2}/2) in Equation (9); it would be better to write the following Equation (10) as prevailing in laminar flows of anticyclones.

^{2}/2 + P + ρgh − (Σg

_{t}) = K’

_{t}) able to be released as heat (+Q

_{v}) in vorticity caused by frictional turbulence at surfaces (Figure 1), such as a rough landscape, wind turbines, airfoils, and in the destructive dissipation of tropical cyclones on land. This process amounts to an irreversible loss of work potential as surface heat, although the charging of anticyclones in the atmosphere with radiant heat absorbed by water and other greenhouse gases, such as CO

_{2}and CH

_{4}as illustrated in Figure 1, represents physicochemical work processes influenced by Coriolis effects of the Earth’s different latitudes rotating at different speeds.

^{5}J (0.42 MJ) per cubic meter predicted in Table 5 at the eye of a tropical cyclone is only 5% greater than the total entropic heat content. The vortical inertial energy shown in Table 3 at a wind speed of 15 m per sec needs only 3.4 × 10

^{3}J of quanta per cubic meter, just over one-thousandth the energy stored in air warmed from absolute zero. From this viewpoint, storing extra energy in air as vortical energy on this scale is unsurprising. If experiments show this prediction is true, this will have important consequences for climate science.

## 5. Conclusions

^{2}) experienced in the correlated molecular motions. These vortical quanta or Gibbs fields must be calculated in the same way as internal motions, including translation, but should be related to coherent torques rather than absolute temperature in degrees Kelvin. This field of impulsive energy traveling within vortical systems at the speed of light provides a new reservoir of energy from work processes, just as valid as for molecules heated as gases as in the Carnot cycle. This new understanding of these reservoirs for heat or work is predicted to be important in tropospheric climate science as it involves reversibility.

- The propositions offered here for atmospheric science are testable, both theoretically and experimentally. Certainly, the detection of the very long wavelengths of translational quanta proposed in anticyclonic winds or tropical cyclones is challenging, and new technology is needed. However, those fields proposed in the Carnot cycle are in the microwave region and for the N
_{2}column in the atmosphere are only an order of magnitude longer. There is an acknowledged dearth of efficient detectors for wavelengths greater than far infrared or microwaves. - A new method to estimate vortical energy is presented based on the action or quantum state of molecules. This is important as it provides a better understanding of how the Earth’s surface is heated as measured by local temperature, how wind power for wind turbines is sustained independent of kinetic energy, and how the destructive power of tropical cyclones is generated from the heat of vaporization of tropical sea water. Far more energy is stored as vortical energy than currently assumed based on laboratory measurements of heat capacity supporting internal energy. This means that heat released from turbulence caused by collisions of air masses or released by wind farms must be considered.
- Practical consequences from the separate analyses conducted in this article include (i) a new means to estimate the lapse rate in the atmosphere that can inform climate models; (ii) a method to model heat absorption in the troposphere by greenhouse gases for recycling (Figure 1) by driving vortical friction at the surface boundary layer; (iii) a new means to predict turbulent heat production downwind of wind turbines, suitable for future field studies; and (iv) a better understanding of the heat-work cycle involved in tropical cyclones, with heat radiated by condensation of water at the eyewall powering the vortical action of the cyclone. All of these findings offer opportunities for new means of gathering information by specific testing technologies.
- The quantum field hypothesis challenges the common opinion that heat is no more than the inertial motion of molecules. Instead, that molecular inertia and motion are sustained by field energy is confirmed, consistent with the density of activated quantum states as in Table 1. The smaller magnitude of translational quanta presents a difficulty for confirmation as there are few spectrometers capable of measuring the intensity of radiation at these long wavelengths. This gap in current technology is likely to be overcome in the future.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Glossary

@ | action or Jdϕ or Iωdϕ where I is the moment of inertia (J.s) |

ϕ | indicates an angle, the ratio of circumference divided by radius expressed as radians or degrees (1 radian ≡ 57.296 degrees or 2π ≡ 360 degrees |

dϕ | an infinitesimal variation in angular motion |

ω | ≡dϕ/dt or Ω, differentiating the rate of change in angular motion with time |

mr | inertia, expressed as mass modified by radius (r) (kg.m) |

mr^{2} | moment of inertia = I |

mr^{2}ω | angular momentum or intensity of action (J.s) |

mr^{2}ω^{2} | energy or vis viva as torque (mv^{2} ≡ T) or twice the kinetic energy (J) |

mr^{2}ω^{3} | power as energy per unit time ≡ TΩ (J/s or W) |

1/r | curvature requiring infinite radius to achieve straight line motion or zero |

J | energy as Joules, the work undertaken when 1 Newton acts over 1 m at constant radius from a center of force |

J | ≡mr^{2}ω or angular momentum (J.s) |

S | action as ∫(T − V)dt, the time integral of the Lagrangian L, the difference between kinetic and potential energy with time (J.s) |

S | entropy or energy per unit of temperature (J/T) |

## References

- Kennedy, I.; Geering, H.; Rose, M.; Crossan, A. A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action. Entropy
**2019**, 21, 454. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kennedy, I.R.; Hodzic, M. Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle. Entropy
**2021**, 23, 860. [Google Scholar] [CrossRef] [PubMed] - Planck, M. The Theory of Heat Radiation; Dover Publications: New York, NY, USA, 1913. [Google Scholar]
- Carnot, S. Réflexions sur la Puissance Motrice du feu et sur les Machines Propres a Developer Cette Puissance; Annales Scientifique de L’ecole Normale Superiere 2e Serie; Carnot, M.H., Ed.; Chez Bachelier: Paris, France, 1872. [Google Scholar]
- Einstein, A. On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. Ann. Phys.
**1905**, 17, 549. [Google Scholar] [CrossRef] [Green Version] - Kennedy, I.R.; Hodzic, M. Partitioning Entropy with Action Mechanics: Predicting Chemical Reaction Rates and Gaseous Equilibria of Reactions of Hydrogen from Molecular Properties. Entropy
**2021**, 23, 1056. [Google Scholar] [CrossRef] [PubMed] - Kennedy, I.R. Computation of planetary atmospheres by action mechanics using temperature gradients consistent with the virial theorem. Int. J. Energy Environ.
**2015**, 9, 129–146. [Google Scholar] - Kennedy, I.R.; Hodzic, M.; Crossan, A.N.; Acharige, N.; Runcie, J. A new method for estimating maximum power from wind turbines: A fundamental Newtonian approach. arXiv
**2021**, arXiv:2110.15117. [Google Scholar] - Feynman, R.P. The principle of least action. In The Feynman Lectures on Physics; California Institute of Technology: Pasadena, CA, USA, 2010; Chapter 29; Volume II. [Google Scholar]
- Kennedy, I.R. Action in Ecosystems: Biothermodyamics for Sustainability; Research Studies Press: Baldock, UK; John Wiley: Baldock, UK, 2001. [Google Scholar]
- Kiehl, J.T.; Trenberth, K.E. Earth’s annual global mean energy budget. Bull. Amer. Meteor. Soc.
**1997**, 78, 197–208. [Google Scholar] [CrossRef] - Tatartchenko, V.; Liu, Y.; Chen, W.; Smirnov, P. Infrared characteristic radiation of water condensation and freezing in connection with atmospheric phenomena; Part 3: Experimental data. Earth-Sci. Rev.
**2012**, 114, 218–223. [Google Scholar] [CrossRef] - Montgomery, M.; Smith, R. Paradigms for tropical cyclone intensification. J. South. Hemisphere Earth Syst. Sci.
**2014**, 64, 37–66. [Google Scholar] [CrossRef] - Emanuel, K.A. Some Aspects of Hurricane Inner-Core Dynamics and Energetics. J. Atmos. Sci.
**1997**, 54, 1014–1026. [Google Scholar] [CrossRef] - Emanuel, K.A. Tropical cyclones. Anu. Rev. Earth Planet Sci.
**2003**, 31, 75–104. [Google Scholar] [CrossRef] - Popper, K.R. Conjectures and Refutations: The Growth of Scientific Knowledge; Routledge & Kegan Paul: London, UK, 1963. [Google Scholar]
- Field, M.J.; Bash, P.A.; Karplus, M. A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations. J. Comput. Chem.
**1990**, 11, 700–733. [Google Scholar] [CrossRef] - Santos, L.D.A.; Prandi, I.G.; Ramalho, T.C. Could Quantum Mechanical Properties Be Reflected on Classical Molecular Dynamics? The Case of Halogenated Organic Compounds of Biological Interest. Front. Chem.
**2019**, 7, 848. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Koehl, P.; Levitt, M. A brighter future for protein structure prediction. Nat. Struct. Biol.
**1999**, 6, 108–111. [Google Scholar] [CrossRef] [PubMed] - Bartels, C. Analyzing biased Monte Carlo and molecular dynamics simulations. Chem. Phys. Lett.
**2000**, 331, 446–454. [Google Scholar] [CrossRef] - Feynman, R. QED: The Strange Theory of Light and Matter. Leonardo
**1991**, 24, 493. [Google Scholar] [CrossRef] - Berkowitz, R. Macroscopic systems can be controllably entangled and limitlessly measured. Phys. Today
**2021**, 74, 16–18. [Google Scholar] [CrossRef] - Ockeloen-Korppi, C.F.; Damskägg, E.; Pirkkalainen, J.-M.; Asjad, M.; Clerk, A.A.; Massel, F.; Woolley, M.J.; Sillanpää, M.A. Stabilized entanglement of massive mechanical oscillators. Nature
**2018**, 556, 478–482. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Global model for the role of vortical action in anticyclones and cyclones, coupled with the Kiehl–Trenberth heat budget [11]. Vortical energy quanta (Σhv) for warming the Earth’s surface are calculated as shown, supporting the vortical action of molecules per m

^{3}circulating around the higher-pressure center, and acting as rotational winds of low curvature in the plane of the Earth’s surface. Note that all symbols (m, s, e, c, a, g, T, h, o, i) or their upper-case versions are specific to this figure.

**Figure 2.**Vortical action (@

_{w}) and entropy in anticyclones. Action for a cubic meter of wind molecules is computed to generate vortical Gibbs energy at r cm from the origin. The associated quantum field has a wavelength λ so that c/V is of the same order as λ/r.

**Figure 3.**Contrasts in function of balanced air foil pairs and wind turbine blade aerodynamics. Air behind the foils moving at forward speed V is depleted, equivalent to the volume of air deflected under the wings (ASinθV/sec, A area and θ angle of wind deflection) and replenished immediately by the drag from air dropping from above. Air in front of the rotor blades is compressed by rotation, exerting a back torque and decreasing wind power, particularly at a longer radius (R). Air foils are normally horizontally balanced in flight, preventing rotation or rolling.

**Figure 4.**Cyclone radius shown in km (x-axis) of a tropical cyclone at 17° S latitude, showing greatest intensity to the southwest and accentuated by the Coriolis effect deflecting the cyclone to the southwest. Wind speed (y-axis m/s, see Table 5) shown in grey peaks at the eyewall; the kinetic energy pressure or torque pressure in blue appears minor, but its intensity is maintained by the much larger Gibbs field energy content.

**Table 1.**Action thermodynamics of the Carnot cycle for working fluids of argon and nitrogen molecules.

Property | Stage 1 | Stage 2 | Stage 3 | Stage 4 |
---|---|---|---|---|

Kelvin temperature | 640–640 | 640–288 | 288–288 | 288–640 |

Argon (Ar) | Isothermal | Isentropic | Isothermal | Isentropic |

Radius (a/2 = r, m) | 6.410895 × 10^{−10} | 8.947125 × 10^{−10} | 13.337586 × 10^{−10} | 9.556798 × 10^{−10} |

Pressure (kT/a^{3}, J/m^{3}) | 4.191891 × 10^{6} | 1.542111 × 10^{6} | 0.2094820 × 10^{6} | 0.5694312 × 10^{6} |

Translational action (@_{t}, J.s) | 12.43697 × 10^{−33} | 17.35719 × 10^{−33} | 17.35719 × 10^{−33} | 12.43697 × 10^{−33} |

Mean quantum number (n_{t =} @_{t},/) | 117.932 | 164.587 | 164.587 | 117.932 |

Negative Gibbs energy (−g_{t,} J) | 12.6446 × 10^{−20} (a) | 13.5282 × 10^{−20} (b’) | 6.0877 × 10^{−20} (a’) | 5.6901 × 10^{−20} (b) |

Mean quantum (hv, J) | 1.07220 × 10^{−21} | 0.82195 × 10^{−21} | 0.36988 × 10^{−21} | 0.48249 × 10^{−21} |

Energy density (g_{t}/a^{3}, J/m^{3}) | 5.998728 × 10^{7} | 2.361020 × 10^{7} | 0.320724 × 10^{7} | 0.814874 × 10^{7} |

Quantum frequency (v, Hz) | 1.61812 × 10^{12} | 1.24045 × 10^{12} | 0.55820 × 10^{12} | 0.72815 × 10^{12} |

Wavelength (m) | 1.85272 × 10^{−4} | 2.41680 × 10^{−4} | 5.37066 × 10^{−4} | 4.11716 × 10^{−4} |

λ/2πr (quanta/molecular) | 4.59951 × 10^{4} | 4.29909 × 10^{4} | 6.40870 × 10^{4} | 6.85654 × 10^{4} |

Molecular frequency (ω) | 9.81843 × 10^{11} | 7.03521 × 10^{11} | 3.16585 × 10^{11} | 4.41829 × 10^{11} |

Ratio (ν/ω) | 1.64804 | 1.76321 | 1.76321 | 1.64804 |

Pressure ratio (g_{t}/kT) | 14.3103 | 15.3103 | 15.3103 | 14.3103 |

nh/λmv × 10^{−5} | 1.0024 c/v = 4.77 × 10^{5} | 1.0732 c/v = 4.77 × 10^{5} | 1.0732 c/v = 7.1 × 10^{5} | 1.0024 c/v = 7.1 × 10^{5} |

Nitrogen (N_{2}) translational | ||||

Radius (a/2= r, m) | 6.410895 × 10^{−10} | 8.947125 × 10^{−10} | 17.40496 × 10^{−10} | 12.47120 × 10^{−10} |

Pressure (kT/a^{3}, J/m^{3}) | 4.191891 × 10^{6} | 1.542111 × 10^{6} | 0.942669 × 10^{5} | 2.56244 × 10^{5} |

Translational action (@_{t}, J.s) | 10.40552 × 10^{−33} | 14.52207 × 10^{−33} | 18.95066 × 10^{−33} | 13.5787 × 10^{−33} |

Mean quantum number (n_{t}) | 98.669 | 137.703 | 179.697 | 128.758 |

Negative Gibbs energy (−g_{t}, J) | 12.1719 × 10^{−20} | 13.0555 × 10^{−20} | 6.1925 × 10^{−20} | 5.79484 × 10^{−20} |

Mean quantum (hv, J) | 1.23361 × 10^{−21} (a) | 0.94809 × 10^{−21} (b’) | 0.34461 × 10^{−21} (a’) | 0.45006 × 10^{−21} (b) |

Energy density (g_{t}/a^{3}, J/m^{3}) | 5.774446 × 10^{7} | 2.278515 × 10^{7} | 0.146810 × 10^{7} | 0.37345 × 10^{7} |

Quantum frequency (v, Hz) | 1.86172 × 10^{12} | 1.43082 × 10^{12} | 0.52007 × 10^{12} | 0.67921 × 10^{12} |

Wavelength (m) | 1.61030 × 10^{−4} | 2.09525 × 10^{−4} | 5.76450 × 10^{−4} | 4.41385 × 10^{−4} |

λ/2πr (quanta/molecular) | 3.99768 × 10^{4} | 3.72712 × 10^{4} | 5.27119 × 10^{4} | 5.04682 × 10^{4} |

Molecular frequency (ω) | 11.73527 × 10^{11} | 8.40869 × 10^{11} | 2.89965 × 10^{11} | 4.04678 × 10^{11} |

Ratio (ν/ω) | 1.58664 | 1.70159 | 1.79355 | 1.67839 |

Pressure ratio (g_{t}/kT) | 13.77530 | 14.77530 | 15.57381 | 14.57381 |

nh/λmv × 10^{−5} | 1.1541 c/v= 3.99 × 10^{5} | 1.2379 c/v= 3.99 × 10^{5} | 0.8753 c/v = 1.14 × 10^{5} | 0.8191 c/v = 1.1 × 10^{5} |

Nitrogen rotational | ||||

Negative rotational Gibbs energy (−g_{r}, J) | 4.1575 × 10^{−20} (a) | 4.1575 × 10^{−20} (b’) | 1.5534 × 10^{−20} (a’) | 1.5534 × 10^{−20} (b) |

Mean quantum number (j_{r}) | 10.513 | 10.513 | 7.052 | 7.052 |

Mean quantum (hv, J) | 3.9547 × 10^{−21} | 3.9547 × 10^{−21} | 2.20268 × 10^{−21} | 2.20268 × 10^{−21} |

Energy density (g_{t}/a^{3}, J/m^{3}) | 1.97236 × 10^{7} | 7.28400 × 10^{6} | 0.36828 × 10^{6} | 1.00107 × 10^{6} |

Frequency (v, Hz) | 5.96831 × 10^{12} | 5.96831 × 10^{12} | 3.32421 × 10^{12} | 3.32421 × 10^{12} |

Wavelength (m) | 5.02308 × 10^{−5} | 5.02308 × 10^{−5} | 9.01846^{−5} | 9.01846 × 10^{−5} |

λ/2πr | 1.24702 × 10^{4} | 8.93525 × 10^{3} | 8.24669 × 10^{3} | 1.15092 × 10^{4} |

n_{t}^{3} × j_{r}^{2} | 1.0616259 × 10^{8} | 2.885798 × 10^{8} | 2.885798 × 10^{8} | 1.0616259 × 10^{7} |

**Table 2.**Tropospheric variation with altitude in molar entropy (S) and Gibbs function (G/T) for the major tropospheric gas N

_{2}(78.04%) in the Model 6 US standard reference atmosphere.

Alt km | Temperature K | Pressure Atm | S_{t}J/mol/K | Mean Level n_{t} | ST Trans. kJ/mol | Mean hν(×10 ^{−22} J)Quanta | Sr J/mol/K | Mean Level n _{r} | Mean (×10^{−21} J)Quanta hν | Total (St + Sr)T kJ/mol | Gibbs G kJ/mol |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 288.2 | 1.000 | 151.76 | 190.69 | 43.74 | 3.809 | 40.92 | 10.05 | 1.94900 | 55.556 | −61.225 |

1 | 281.7 | 0.886 | 152.28 | 194.71 | 42.90 | 3.659 | 40.73 | 9.94 | 1.91743 | 54.370 | −60.284 |

2 | 275.2 | 0.785 | 152.81 | 198.93 | 42.05 | 3.511 | 40.54 | 9.82 | 1.88723 | 53.210 | −59.316 |

3 | 268.7 | 0.692 | 153.36 | 203.34 | 41.20 | 3.446 | 40.34 | 9.70 | 1.85625 | 52.047 | −58.340 |

4 | 262.2 | 0.609 | 153.91 | 207.92 | 40.36 | 3.223 | 40.14 | 9.59 | 1.82303 | 50.880 | −57.365 |

5 | 255.7 | 0.534 | 154.48 | 212.73 | 39.50 | 3.083 | 39.93 | 9.47 | 1.79095 | 49.771 | −56.385 |

6 | 249.2 | 0.466 | 155.08 | 217.89 | 38.65 | 2.946 | 39.71 | 9.35 | 1.75808 | 48.544 | −55.405 |

7 | 242.7 | 0.401 | 155.78 | 224.09 | 37.81 | 2.802 | 39.49 | 9.22 | 1.72675 | 47.394 | −54.444 |

8 | 236.2 | 0.352 | 156.36 | 228.80 | 36.93 | 2.680 | 39.27 | 9.10 | 1.69318 | 46.194 | −53.475 |

9 | 229.7 | 0.302 | 157.00 | 235.26 | 36.06 | 2.545 | 39.04 | 8.97 | 1.66066 | 45.028 | −52.458 |

10 | 223.3 | 0.262 | 157.59 | 240.92 | 35.19 | 2.426 | 38.80 | 8.85 | 1.62623 | 43.854 | −51.471 |

11 | 216.8 | 0.224 | 158.28 | 247.67 | 34.59 | 2.319 | 38.56 | 8.72 | 1.59251 | 42.672 | −50.479 |

12 | 216.7 | 0.192 | 159.55 | 260.63 | 34.57 | 2.203 | 38.55 | 8.71 | 1.58327 | 42.928 | −50.737 |

^{7}ergs) is available as Table S2 Entropy8/Cal in Supplementary Materials. S

_{t}and S

_{r}are entropies per mole for translation and rotation, respectively.

Wind Speed V (m/s) | Kinetic Energy/s 83 m Diam J | Kinetic Energy /Blade-Area/s J | Vortical Pressure, J/m ^{3} | Vortical Power for Blade Area Watts | Power Estimated by Radial Action Model Watts |
---|---|---|---|---|---|

At λ = 9, pitch θ = 55° | |||||

5.0 | 0.21670 × 10^{6} | 8.4066 × 10^{3} | 0.36107 × 10^{3} | 0.33038 × 10^{6} | 0.031168 × 10^{6} |

10.0 | 1.7336 × 10^{6} | 6.7253 × 10^{4} | 1.47258 × 10^{3} | 2.69482 × 10^{6} | 0.40541 × 10^{6} |

15.0 | 5.8509 × 10^{6} | 2.2698 × 10^{5} | 3.35055 × 10^{3} | 9.19727 × 10^{6} | 1.54381 × 10^{6} |

20.0 | 1.3869 × 10^{7} | 5.3802 × 10^{5} | 6.00353 × 10^{3} | 21.9729 × 10^{6} | 3.86798 × 10^{6} |

Wind Speed (m s ^{−1}) | Vortical Action (mrv/2 =@_{v})/Molecule J.s, ×10^{19} | Quantum Number n _{vor} ×10 ^{−15} | 1-D Torque mv ^{2}/Mole-Cule ×10^{−24}J | Vortical Energy /Molecule [(mv ^{2})ln(n_{vor}), ×10^{−23} J | Vortical Energy J/m ^{3} | Vortical Wavelength ×10 ^{−12} m | Kinetic Energy J/m ^{3} | Ratio Vortical/ Kinetic Energy |
---|---|---|---|---|---|---|---|---|

5.0 | 1.2108 | 1.14812 | 1.2108 | 4.2812 | 1062.024 | 5.3272 | 15.313 | 69.355 |

10.0 | 2.4215 | 2.29615 | 4.8430 | 17.1297 | 4332.826 | 2.6627 | 61.250 | 70.737 |

15.0 | 3.6823 | 3.49168 | 10.8968 | 38.998 | 9864.433 | 1.7786 | 137.813 | 71.758 |

20.0 | 4.8430 | 4.09488 | 19.372 | 69.862 | 17670.951 | 1.1643 | 245.000 | 72.127 |

**Table 5.**Vortical entropic energy from infrared radiation of water condensation at the convective eyewall of a tropical cyclone.

Radius km | R | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500 |
---|---|---|---|---|---|---|---|---|---|---|---|

Speed m/sec | rω | 100 | 50 | 33.33 | 25 | 20 | 16.67 | 14.29 | 12.5 | 11.11 | 10 |

@_{v} J-sec × 10^{15} | n_{v} = mr^{2}ω/2ħ | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 | 1.481 |

Torque/molecule Joules × 10 ^{−24} | mv^{2} = mr^{2}ω^{2} | 484.3 | 121.1 | 53.81 | 30.27 | 19.372 | 13.453 | 9.884 | 7.567 | 5.979 | 4.843 |

−Gibbs/molecule Joules × 10 ^{−22} | mv^{2}ln[n_{v}] | 167.940 | 41.985 | 18.660 | 10.496 | 6.717 | 4.665 | 3.427 | 2.624 | 2.073 | 1.679 |

Kinetic/molecule J/m ^{3} × 10^{−3} | p = 1.5kT/a^{3} | 6.125 | 1.531 | 0.681 | 0.383 | 0.245 | 0.170 | 0.125 | 0.096 | 0.076 | 0.061 |

−Gibbs J/m^{3} × 10^{5} | p = J/m^{3} | 4.247 | 1.062 | 0.472 | 0.265 | 0.170 | 0.118 | 0.087 | 0.065 | 0.052 | 0.046 |

Ratio pressures | Gibbs/kinetic | 69.3 | 69.4 | 69.3 | 69.2 | 69.4 | 69.4 | 69.6 | 68.7 | 68.4 | 75.4 |

^{5}J/m

^{3}; the integrated thermal input for 25 mm evaporation a day sustains a 500 km radius cyclone 10 km high containing 8.794743 × 10

^{19}J of energy, approximately twice the daily input; see Table S4 (Tropcyc2/Cal program) in Supplementary Materials for program.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kennedy, I.R.; Hodzic, M.
Applying the Action Principle of Classical Mechanics to the Thermodynamics of the Troposphere. *Appl. Mech.* **2023**, *4*, 729-751.
https://doi.org/10.3390/applmech4020037

**AMA Style**

Kennedy IR, Hodzic M.
Applying the Action Principle of Classical Mechanics to the Thermodynamics of the Troposphere. *Applied Mechanics*. 2023; 4(2):729-751.
https://doi.org/10.3390/applmech4020037

**Chicago/Turabian Style**

Kennedy, Ivan R., and Migdat Hodzic.
2023. "Applying the Action Principle of Classical Mechanics to the Thermodynamics of the Troposphere" *Applied Mechanics* 4, no. 2: 729-751.
https://doi.org/10.3390/applmech4020037