# An Entropy Generation Rate Model for Tropospheric Behavior That Includes Cloud Evolution

## Abstract

**:**

## 1. Introduction

#### 1.1. The Earth’s Atmosphere

_{2}, O

_{2}, and aerosols. The average depth (or the tropopause level) of the troposphere is ~17 km (about 10.56 mi) in the middle latitudes, shallower at the cold poles ~7 km, and higher at the equator ~20 km. The troposphere is relatively shallow compared to the other layers of the atmosphere. However, the troposphere contains all atmospheric clouds except for a rare polar stratospheric cloud. The Earth’s surface temperature and surface texture strongly influence winds in the lower levels of the troposphere. Figure 1 illustrates the critical phenomena in the Earth’s atmosphere relevant to this article. Figure 1a shows the troposphere dimensions and temperature gradients in the atmospheric layers above the Earth’s surface. Figure 1b shows the typical wind patterns and radiation inlet and outlet into the troposphere. Figure 1c shows typical clouds observed in the troposphere. Figure 1d shows a typical temperature profile in the troposphere and provides a pictorial definition of some of the “lapse rate” temperature profiles discussed in the article. This article explores the relationship between the entropy generation rate in the troposphere and cloud morphology (patterns) for understanding weather. The caption of Figure 1 defines some of the parameters that are important for the model to follow.

**Figure 1.**The figure illustrates all the major phenomena in the Earth’s atmosphere that will be important to the model and the discussions to follow. (

**a**) The Earth’s atmosphere and temperature profiles across segments of the atmosphere. The troposphere is a small segment of the Earth’s atmosphere but contains almost all the weather and clouds; (

**b**) the typical equatorial, tropical, and polar wind patterns in the northern hemisphere; (

**c**) common types of clouds and their altitudes [1], (

**d**) the typical temperature and various “lapse rates” in the troposphere—close to the Earth’s surface, lapse rate diurnal inversions are often noted, as shown in the schematic [1,2]. Definitions: A

**cloud**is a mass of tiny water drops or ice crystals that float in the air above Earth. In this article, clouds are treated as diffuse interfaces.

**Condensed water**: the liquid that is precipitated from a supersaturated H

_{2}O/air gas.

**Cloud water content**: the cloud liquid water content is a measure of the total liquid water contained in a cloud in a vertical column of the atmosphere. It does not include solid water (snow, ice).

**Precipitable water**is the amount of water potentially available in the atmosphere for precipitation, usually measured in a vertical column that extends from the Earth’s surface to the upper edge of the troposphere.

_{E}/dt~0.01 to 0.018 K/year or (3.17 to 5.7) × 10

^{−10}K/s [3,4,5,6]. The absorption of infrared radiation by tropospheric gases is significant to the global energy balance. The increasing number of molecules in the Earth’s atmosphere that have three or more atoms, like water (H

_{2}O), methane (CH

_{4}), carbon dioxide (CO

_{2}), and nitrous oxides (NO

_{x}), absorb and transmit in the infrared spectrum, which leads to the Earth’s surface radiation being trapped, thus increasing the Earth’s surface and tropospheric warming. The average warming is accelerating for several reasons, including a lowering of the albedo—a measure of the proportion of incident radiation from outer space and the sun reflected by a surface. Since the greenhouse effect impacts clouds and ice melting, a change in albedo can contribute to overall warming temperatures. This article additionally considers emissivity changes in various cloud formations that impact the energy and entropy balance in the troposphere.

#### 1.2. Entropy Creation and Exchange

#### 1.3. Atmospheric Stability

#### 1.4. Cloud Evolution and Intense Weather

_{2}O, the clouds can stretch or connect horizontally or grow vertically. Horizontal clouds are not associated with external updrafts (see Table 1). Such clouds form at any altitude. When several nuclei are present, and a strong driving force for nucleation exists, the horizontal clouds form at low altitudes; otherwise, they form at mid to high altitudes (Figure 1b). Mid- and high-level thin clouds aid warming. Regardless of external updrafts or heterogeneous nuclei, the condensates in clouds, experience some undercooling before precipitation. The volume change and rapid heat release generated during the recalescence process cause local updrafts. When an updraft is possible, i.e., vertical movement is possible in the clouds, the clouds assume a cumulus character. The transition from flat to cumulus character is addressed in this article. It is known that horizontally layered stratus clouds are not intense-weather-causing phenomena unless instabilities manifest in the vertical direction. Cumulus clouds that only show slight vertical growth are associated with fair weather. When cumulus clouds begin to grow and resemble a head of cauliflower, they are called cumulus congestus, or swelling cumulus, which can further lead to towering cumulus. Vertical clouds are intense-weather-causing clouds. These clouds can develop into a giant cumulonimbus, i.e., thunderstorm clouds. The formation of rapidly growing vertical clouds that cause intense weather is favored by an unstable atmosphere, i.e., where strong updrafts are prevalent. Vertical clouds of the kind shown in Figure 1b are often the cause of intense weather with thunderstorms, hail, and lightning. Such clouds are studied in this article with a previously described entropy rate diffuse-interface model [17,45].

_{2}O particles with a post-nucleation droplet size of about 1 micrometer (micron). These tiny cloud droplets tend to be carried along with the vertically moving air because the flow required for suspension of even a 30-micron droplet is only 0.02 m/s—far lesser than the typical cloud updraft velocity of about ten m/s. The total water mixing ratio and entropy are conserved by parcels that have nucleated but are still in an updraft. The initially tiny starter cloud water droplets can coalesce into larger sizes greater than ~1 mm (about 0.04 in), which makes them challenging to suspend with the updraft velocity. Updrafts cannot carry larger droplets, e.g., a ~3 mm (about 0.12 in) water-particle diameter; the updraft must exceed 20 m/s to keep it suspended. Consequently, the large particles fall because of gravity. Maritime clouds have larger water-droplet sizes compared with land (continental) clouds. The droplet distribution typically varies between a few and 50 microns in clouds. As the water droplets and ice particles rise and coalesce, the droplets and ice crystals become larger and can no longer be supported by the cloud updraft. These large water droplets or hail fall to the ground, often with rapid downdrafts.

^{3}of water evaporates from the land and ocean surface annually, remaining for about ten days in the atmosphere before falling back to the surface as rain or snow/ice. The estimates are that 86% of global evaporation and 78% of global precipitation occur over the oceans; thus, a net water transfer occurs over land regions (the water is returned by riverways to the oceans). However, climate change is likely causing parts of the water cycle to speed up and change the return pathways as warming global temperatures increase the evaporation rate worldwide. On average, more evaporation causes more precipitation. Climate change affects the world’s water in complex ways, from unpredictable rainfall patterns to shrinking ice formations, rising sea levels, severe floods, and droughts.

#### 1.5. Lapse Rates

_{p}), where g is the gravitational constant and C

_{p}is the specific heat

_{.}This is the lapse rate of unsaturated air and is roughly linear ~−9.76 K/km with altitude. When the temperature falls below the dew point, subject to overcoming nucleation difficulties, cloud formation is observed in the troposphere. Saturated air is at the dew-point temperature (a condition that depends on the water content and pressure). The local moist adiabatic lapse rate (MALR) is the lapse rate of saturated air with a condensed phase. The saturation and lower-than-saturation lapse rate (SLR) and the MAPR are seemingly similar for saturated and near-saturated conditions, but the SLR can extend below the cloud base. The dew point also has a lapse rate. It tracks the density profile of the air masses and is typically about ~−1.8 K/km. However, closer to the Earth’s surface, the dew point may also show a positive lapse rate. Some typical lapse rate illustrations are shown in Figure 1c.

#### 1.6. PBL

## 2. Tropospheric Entropy Generation Rate

_{Earth}.h. For the model, the troposphere base and top are considered equal making the control volume base area A

_{Earth}equal to the Earth’s actual surface area and height, h, equal to the average height of the tropopause. For a control volume that begins at the tropopause and extends to the Earth’s surface, it is possible to write an entropy (S, J/Kg.K) balance as:

**S**

_{cv}= (S

_{in}− S

_{out}) +

**S**

_{gen}

^{3}), C

_{p}(J/kg.K), are the troposphere’s average density and specific heat (at constant pressure), respectively, t is time, and CV is the control volume. T is the temperature variable, T

_{av}is the average temperature of the control volume, and T

_{E}is the average temperature of the Earth’s surface. The

**s**

_{gen}is the entropy generated per unit volume. d

**s**

_{gen}/dt is the entropy generation rate per unit volume that captures all the entropy-generating features from the mass and temperature gradients and phase changes in the control volume [13,17]. The incoming entropy is dS

_{in}/dt.Δt + entropy transfer from the Earth’s interior via the temperature gradient plus the entropy added by the additional water vapor minus entropy lost from the troposphere by condensing water or ice that falls back to Earth. This is balanced by the entropy accumulation and the entropy out plus the entropy generated. The entropy accumulated and generated in the control volume (CV) is given by:

**S**

_{cv}= AhρC

_{p}d(ln(T

_{av}))

**S**

_{gen}= Ah(d

**s**

_{gen}/dt)Δt

_{α}is the average emissivity of the Earth.

_{in}= A

_{Earth}ψ(ds

_{in}/dt)Δt

^{−5}. Without clouds, the entropy leaving the troposphere is

_{out}= A

_{Earth}(ds

_{out}/dt)Δt = (4ε

_{α}σ A

_{Earth}T

^{3}/3)Δt + (4σT

_{sun}

^{3}/3)κψΔt + dS

_{Earth-core}/dt

^{−8}J/m

^{2}K

^{4}s). An assumption made is that the entropy transfer with solid Earth is small, for electrical jets (lightning). In the presence of clouds, the outgoing entropy by radiation is:

_{out}= (4A

_{Earth}Φε

_{χ}σT

_{c}

^{3}/3)Δt + (4σA

_{Earth}(1 − Φ)ε

_{α}T

_{E}

^{3}/3)Δt + (4σT

_{sun}

^{3}/3)κ

_{1}ψΔt + dS

_{Earth-core}/dt

_{1}is the catch-all average percentage reflected by the Earth and clouds not otherwise captured, e.g., the radiation from Earth and from between clouds that are not mitigated by clouds.

_{ϕ}, and Tc represent the effective cloud coverage, the average cloud emissivity, and the average cloud temperature radiating out of the troposphere, respectively, which establishes the entropy transport out of the tropospheric control volume from clouds. The term ε

_{ϕ}also encompasses the influence of the GHG molecules on emissivity.

_{Earth-core}/dt = [KA

_{Earth}(dT

_{E-core}/dx)

^{2}/T

^{2}] is the incoming entropy inside the Earth to the tropospheric control volume. Taking the average thermal conductivity of the soil, even with a high gradient of 30 K/km, gives dS

_{Earth-core}/dt as a small number compared to the radiative terms, so it will be ignored.

_{p}(ln(T)) = (4σψT

_{sun}

^{3}/3)Δt − (4ε

_{α}σT

_{E}

^{3}/3)Δt − (4σT

_{sun}

^{3}/3)κψΔt + hd

**s**

_{gen}/dtΔt + dΔδ/dtΔt − ΛΔt

_{av}is the average temperature of the troposphere. Λ is the entropy-loss term associated with the catch-all electron jet (like lightning) that crosses the boundaries of the tropospheric control volume (this term will be considered to be small).

_{av}/dt = ~dT

_{cv}/dt = ~dT

_{E}/dt for the global scale. Noting that ds

_{in}/dt is a constant (not dependent on the Earth or cloud temperature), the time derivative of Equation (6) yields

_{p}(ln(T

_{av}))V + hρC

_{p}(d(ln(T

_{av}))/dt) = hd

**s**

_{gen}/dt − (4σε

_{α}T

_{E}

^{3})/3 + dΔδ/dt − Λ

_{p}(ln(T

_{av}))V + hρC

_{p}(d(ln(T

_{av}))/dt) = hd

**s**

_{gen}/dt − (4Φε

_{χ}σT

_{c}

^{3})/3 − (4σ(1 − Φ)ε

_{α}T

_{E}

^{3})/3 + dΔδ/dt − Λ

_{χ}are the fraction cloud coverage and emissivity of the clouds, respectively.

^{2}T/dt

^{2}= 0. Note that ρC

_{p}(d(T

_{av})/dt)V/T is small compared to the radiation entropy terms. Further assuming (ds

_{gen}/dt) is at an extremum implies that (d

^{2}s

_{gen}/dt

^{2}) is zero (or the entire troposphere is at a quasi-steady state at a maximum entropy production rate), so the above equations may be simplified to:

_{gen}/dt = (ρC

_{p}/T

_{av})d(Tav)/dt + (4σε

_{α}T

_{E}

^{2}/V)dT

_{E}/dt + C

_{av})/dt being small in the simulation without clouds, the term ρC

_{p}/T

_{av}is much smaller than 4σε

_{α}T

^{2}/V and Equation (9a) can be approximated to the following expression.

_{gen}/dt~(4σε

_{α}T

_{E}

^{2}/V)(dT

_{E}/dt)

_{gen}/dt = 5.45 × 10

^{−5}W/m

^{3}K for the current Earth conditions (T

_{E}= 287.5 and dT

_{E}/dt = 5.7 × 10

^{−10}K/s).

_{gen}produced per degree of warming as:

_{gen})/dt)/(d(T

_{E})/dt) = 4σε

_{α}T

_{E}

^{2}/V

^{3}m and the surface area of the Earth is 5.1 × 10

^{14}m

^{2}yields the total entropy generation rate for the troposphere equal to dSgen/dt equal to 5.559 × 10

^{−14}J/(K.s) for the current temperature of 287.5 K (i.e., ds

_{gen}/dt = 5.45 × 10

^{−5}W/m

^{3}K). This entropy generation dSgen/dt calculated with the low complexity model is close to the value of (6.4–6.5) × 10

^{−14}J/(K.s) reported by Wu and Liu [33].

_{gen}/dt~[4Φσε

_{χ}Tc

^{2}dT

_{E}/dt + 4σε

_{α}(1 − Φ)T

^{2}dT

_{E}/dt]/V − (−dΔδ/dt) + dΛ/dt

_{2}concentration, and in 2020, it was 14.6 °C with 370 ppm, which gives the entropy generation rate density per ppm CO

_{2}as approximately equal to 6 × 10

^{−9}J/(m

^{3}.K.s.ppm(CO

_{2})).

**Figure 2.**Entropy rate generation per unit volume vs. Earth’s temperature (K) at the present rate of surface warming and assuming no clouds and dT

_{E}/dt = 0.018 K/year. The Stefan–Boltzmann constant σ = 5.57 × 10

^{−8}W/m

^{2}K

^{4}, emissivity ε

_{α =}0.85, tropospheric expansion velocity V = (1.58 and 3.17) × 10

^{−7}m/s. Cloud formation can increase or mitigate the demand on the entropy generation rate (vertical arrows indicate this).

## 3. Discussions

_{a}increases in the polar regions. Altocumulus clouds have a mean emissivity of about 0.8. The emissivity of high-altitude cirrus clouds ranges from 0 to 1, with a mean of 0.35 [8]. The emissivity of water (oceans) is ~0.95. Thus, new cloud formation over the oceans may not significantly alter or even reduce the entropy production rate demand when compared to a situation without clouds because the emissivity of the underlying water is like that of clouds—unless there is a net loss of entropy from water exchange in the troposphere in the presence of clouds. On the other hand, the emissivity of clouds appears to be larger over land, which, based on Equation (10), would make the presence of clouds increase the demand on the rate of entropy generation. A rigorous sensitivity analysis for the uncertainty in the output of the mathematical model where it can be divided and allocated to different sources of uncertainty in its inputs was not performed, because several terms are dominated by a single physical phenomenon for the complex tropospheric system. Regardless, the sensitivity of Equation (10) to cloud coverage and emissivity change is inferred to be high—this is why the conclusions offered in the article can track both diurnal and long-term changes.

#### 3.1. Entropy Generation in a Cloud without Significant Vertical Development

_{p}(ln(T

_{c(av)})) = (4σψT

_{sun}

^{3}/3)Δt + (4λε

_{α}σT

_{E}

^{3}/3)Δt − (4ε

_{χ}.σT

_{cb}

^{3}/3)Δt − (4ε

_{χ}κ σT

_{c}

^{3}/3)Δt + ξ.d

**s**

_{gen}/dt. Δt − dΔδ/dt

_{c(av)}and Tc are the average and cloud-top cloud temperatures, respectively. The terms ξ, and T

_{cb}are the cloud thickness and the temperature of the cloud base, which for low-lying clouds could be approximated to be the temperature at the PBL height, a constant from known observations. On the RHS, the first term is incoming entropy, the second is outgoing entropy generated by the radiation from the Earth and mitigated by clouds and the tropospheric atmosphere, and the following two terms are the entropy amounts leaving the cloud towards Earth and the stratosphere, respectively. These include the term κ, which considers the reflectivity of radiation between the Earth and cloud bottoms. Here, λ is the correction factor for the pass-through radiation in thin transparent clouds.

_{p}ln(T

_{c(av)})V

_{c}+ hρC

_{p}(d(ln(T

_{avc}))/dt) = ξd

**s**

_{gen}/dt + 4λε

_{α}σT

_{E}

^{3}/3 − (4ε

_{χ}σT

_{cb}

^{3})/3 − (4ε

_{χ}σT

_{c}

^{3})/3 − ξΔf

_{v}ds

_{water}/dt

_{v}ds

_{water}/dt is the rate of entropy loss from the cloud because of rainfall. T

_{avc}is the average cloud temperature. A

_{c}is the cloud area projected normal to the Earth’s surface, r

_{v}is the mixing ratio of water vapor, and Δs

_{water}is the entropy difference from vapor to liquid at the equilibrium transformation temperature at the altitude pressure.

_{c}= d(ξ)/dt = 0, i.e., there is no cloud thickening, and if the second term on the LHS of Equation (12) is small, the entropy generation rate per unit volume is given by

_{gen}/dt = [(4ε

_{χ}σT

_{cb}

^{3}) − (4σψ T

_{sun}

^{3}) + (4ε

_{χ}κ

_{1}σT

_{c}

^{3})] – [(4λε

_{α}σT

_{E}

^{3})]/3ξ

_{gen}/dt~[(8ε

_{χ}σT

_{c}

^{3}− 4ε

_{α}σT

_{E}

^{3})]/3ξ ~ [(8ε

_{χ}σT

_{c(av)}

^{3}− 4ε

_{α}σT

_{E}

^{3})]/3ξ

_{gen}/dt~[(8ε

_{χ}κ

_{1}σT

_{c(av)}

^{3}− 4λε

_{α}σT

_{E}

^{3})]/3ξ + [ξΔf

_{v}ds

_{water}/dt]/ξ

_{c(av)}is approximately ½(T

_{c}+ T

_{cb}).

_{gen}/dt~[[(8ε

_{χ}κ

_{1}σT

_{c}

^{2}− 4λε

_{α}σT

_{E}

^{2})]/3] + [ξΔf

_{v}ds

_{water}/dt] + [ρC

_{p}(ln(T

_{avc}))V

_{c}]SLR/ΔT

_{1}is the average catch-all percentage of radiation the Earth and clouds reflect, λ is the correction factor for the pass-through radiation in thin transparent clouds, and ΔT is the temperature difference between the cloud’s bottom and top and

_{v}is the mixing ratio of water vapor, $g$ is the acceleration due to gravity, C

_{p}is the specific heat at constant pressure of dry air, and r

_{v}is the water-vapor mixing ratio. We can approximate $\mathsf{\Delta}S~\mathsf{\Delta}Sv~\mathsf{\Delta}H/T$ the entropy of vaporization; R is the gas constant for dry air (universal gas constant divided by the molar mass), ɳ is the ratio of the gas constants for dry air and water vapor, $\mathsf{\Delta}H$ is the heat of vaporization, and T is the equilibrium temperature. The subscripts d, v, and l indicate dry air, water vapor, and liquid water, respectively. Equation (15c) reduces to the DALR for dry air, i.e., when r

_{v}= 0, and approximates the MALR at saturation. As the temperatures decrease with altitude, ΔS will correspondingly increase with a drop in temperature, i.e., with altitude, assuming that the heat of vaporization is not a function of temperature. For r

_{v}less than 0.03, this Equation, although nonlinear with ΔS, can make the theoretical SLR less steep and thus increase the thermodynamic driving force for precipitation. The water vapor mixing ratio is in g kg

^{−1}. The water vapor mixing ratio, r

_{v}, is typically at most about 38 g kg

^{−1}or 0.038 kg.kg

^{−1}. A simplified SLR for low saturation that is sometimes employed is −$\mathrm{g}$/C

_{p}(1 − 0.85r

_{v}); however, this simplification does not capture the high undercooling (higher ΔS) that can yield the correct SLR, as shown in Figure 3. Note that this ratio is well above the number two, which is an indicator of a smoothly bounded diffuse interface [17].

**Figure 3.**A plot of SLR (K/km) as a function of ΔS (J/Kg. K). The heat of vaporization ΔH = 2.501 × 10

^{6}Jkg

^{−1}is assumed constant, C

_{p}= 1005 JK

^{−1}kg

^{−1}, C

_{pv}= 1850 JK

^{−1}kg

^{−1}, the gas constant for water vapor R

_{v}= 461 JK

^{−1}kg

^{−1}. The r

_{v}, is the mixing ratio of water vapor (mass ratio). The thick line corresponds to high water content, r

_{v}= 0.038 (3.8%). The thin line is for low water content, r

_{v}= 0.01 (1%).

_{v}, i.e., the water content. Thus, enhanced evaporation and higher tropospheric water content (from the increasing temperature enabled by climate change) can always lead to more condensation. Measurements have shown, however, that the undercooling does not necessarily keep increasing with height above the cloud base but tends to diminish as the T

_{c}level rises toward the tropopause [2]. The difference between SLR at saturation conditions (an approximation for the MAPR) and the dew-point temperature at any altitude is the undercooling (driving force). The higher water content in the troposphere (or sometimes called hydrosphere) increases the precipitation, leading to more water per unit volume that can coalesce and drop to the Earth’s surface.

^{3}and 200 drops/cm

^{3}. Land clouds have much higher droplet concentrations, up to around 900 drops/cm

^{3}. The effective particle radius ratio of condensed water droplets and ice particles in clouds over oceans differs significantly from that of land. For oceans, it is around 14 μm and 25 μm, respectively. In contrast, the two sizes are the same over land, suggesting that the land clouds involve vapor-to-condensed phase transformation occurring at much higher altitudes, thus also suggesting that the entropy generation rate could be much higher with land clouds because of higher undercooling from stronger updrafts, i.e., more entropy generation is possible per unit volume for land clouds.

_{gen}/dt~[[(8ε

_{χ}κσT

_{c}

^{2}− 4ε

_{α}λσT

_{E}

^{2})]/3] + [ξΔf

_{v}ds

_{water}/dt] + [ρC

_{p}ln(T

_{c(av)})V

_{c}]SLR/ΔT

_{gen}/dt~[(ξΔf

_{v}ds

_{water}/dt)SLR/ΔT]

_{water}~S

_{vpi}is the entropy change for transforming vapor into water or ice. Note that the moist lapse rate of near-saturation air is a function of the entropy of condensation, which, if it happens at a lower temperature than the equilibrium at any altitude, will lead to a higher entropy of transformation, provided the enthalpy of transformation is unaffected by the undercooling.

_{E,}and produce rain compared to high stratus with small vertical development or cirrus clouds (ice-containing). Stratus and cirrus clouds are sometimes spread out in patches, with ample sky breaks between them. Stratus clouds are thick, gray clouds that look like fog with a base above the ground. Stratus clouds often produce light, drizzly rain or snow, especially from a nimbostratus cloud. These typically are low-altitude clouds. Clouds may be able to remain stratus-like once they are formed and grow or join other horizontal clouds forming at similar altitudes.

_{sgen}/dt cannot be less than zero. Although the second and third terms in Equation (15d) can influence the entropy generation, the grouping of the first term on the RHS of Equation (15d) must be positive. Cloud types such as status and cirrus with no vertical development are thus limited in their ability to generate entropy. Although horizontal stratus clouds often produce light, drizzly rain or snow, especially when nimbostratus, the thin cloud formations do not easily cool the Earth’s surface, and consequently T

_{c}< T

_{E}.

_{c}~0 i.e., when there is no rain (the role of V

_{c}is made more apparent in the next section). Therefore, for clouds with no vertical development and no change in the rainfall rate, the entropy generation per unit volume rate can only be enhanced in the troposphere by progressively thinning clouds (low ξ). Should they form, cirrus and stratus, i.e., horizontal clouds, must thin (become wispy and disperse) when the demand on the rate of entropy increases because of the warming that these clouds enable. For small entropy generation in horizontal clouds with limited vertical development, the following bifurcation condition at least must be met when there is no rain and if LSR~0:

_{χ}κ

_{1}T

_{c(av)}

^{3}/ε

_{α}λT

_{E}

^{3}> 1

#### 3.2. Entropy Generation in a Cloud with Vertical Development

_{2}O. A mixed layer can form to a height where the static stability of the air forms a barrier to thermally induced upward motion, particularly as the size of the condensed phase increases. This occurs practically daily over the arid areas of the world, where the limit to upward mixing is often the tropopause itself.

_{c}= d(ξ)/dt is non-zero, and if dT

_{cb}/dt = 0, i.e., the cloud base temperatures are constant,

_{gen}/dt~[ξ dΔ(f

_{v}(s

_{vpi})/dt)] + [ρC

_{p}ln(

**T**)V

_{c(av)}_{c}]SLR

**/**ΔT

_{v}, i.e., with altitude.

_{gen}/dt becomes a function of ξ, V

_{c}, and SLR. The loss in work potential (free-energy dissipation) in a vertically unstable cloud is the entropy generation rate multiplied by the average temperature. The cloud thickness ξ is the thickness of the diffuse interface. T

_{av}. dsgen/dt represents the rate of work potential loss for an average value of f

_{v}in the diffuse interface. Following the derivations shown in references [17,45,55,74], the rate can be written as

_{gen}/dt = ∆ρ

_{k}f

_{v}V

_{c}

^{3}/2ξ

^{2}SLR

_{k}is the density difference between the condensed phase and the vapor phase. Alternately, again from references [17,45], it is possible to write the ds

_{gen}/dt (J/m

^{3}.K.s) in the following form:

_{gen}/dt = V

_{c}f

_{v}Δh

_{vl}SLR/T

_{cb}T

_{cT}

_{v}in clouds, SLR, enthalpy of precipitation, and cloud envelope temperatures determine the entropy generation rate per unit volume in clouds.

_{vl}is volumetric, e.g., in the units of J/m

^{3}. The typical values of f

_{v}for various cloud types are shown in Table 1.

_{Earth}h[4Φσε

_{χ}T

_{c}

^{2}dT

_{E}/dt + 4σε

_{α}(1 − Φ)T

_{E}

^{2}dT

_{E}/dt]/V = Acξ[∆ρ

_{k}f

_{v}V

_{c}

^{3}/2ξ

^{2}SLR]

_{Earth}h (4σε

_{α}T

_{E}

^{2}/V)(dT

_{E}/dt) = Acξ[∆ρ

_{k}f

_{v}V

_{c}

^{3}/2ξ

^{2}SLR]

_{c}

^{3}= (2A

_{Earth}h SLR(4σε

_{α}T

_{E}

^{2}/V)(dT

_{E}/dt))/(Ac∆ρ

_{k}f

_{v})

_{c}is the cloud updraft rate. A

_{Earth}and A

_{c}refer to the area of the Earth and cloud, respectively. Plugging in typical values of the surface areas, a cloud Earth/cloud area ratio of 0.66, and the parameters used in Figure 1, with f

_{v}= 3 × 10

^{−6}, SLR = 4 × 10

^{−3}K/m, and a tropospheric height of 20 km, Equation (22) yields

_{c}(m/s)~(0.05ξ)

^{0.33}

_{c}will increase with dT

_{E}/dt (i.e., warming will increase the severity of the weather by enhancing updrafts, which leads to severe weather). Equation (23), plotted in Figure 4, is the calculated updraft velocity from the diffuse-interface formulation. For a near-constant cloud fraction A

_{Earth}/Ac and identical ∆ρ

_{k}and r

_{v}for the cloud as it evolves, any increase in SLR and cloud thickness will increase weather severity, i.e., the cloud can evolve, e.g., into a hail-spewing thundercloud. When T

_{E}decreases because of low-lying fixed-latitude clouds, shielding the sun’s warming, or if the SLR decreases, V

_{c}will decrease. This situation also simulates the final stage of a thunderstorm, where a decrease in the cloud thickness

**ξ**will reduce the entropy rate production. An approximation where the vertically developing V

_{c}> ~0.1 m/s could be a measure for entropy-producing clouds becoming dominant over other entropy-production phenomena like winds. If the entropy-generating high-water-content cloud moves horizontally, typical movement from maritime regions to regions of high surface temperatures, hurricanes (or tornadoes over land) and lightning can manifest. Finally, in the last stage of a heavy rain-or-hail-fall cumulonimbus cloud, the system attempts to approach a new steady state and leaves behind wispy cirrus clouds. The equations for the entropy generation rate from references [17,45] can also be used to extend Equation (19) in the following manner.

_{gen}/dt = V

_{c}f

_{v}Δh

_{vl}SLR/T

_{cb}T

_{cT}= {[(1/SLR)dΔ(f

_{v}(s

_{vpi})/dt)] + ρC

_{p}ln(T

_{avc})V

_{c}/ΔT}SLR

_{c}= [ξ dΔ(f

_{v}(s

_{vpi})/dt) + C

_{p}ln(T

_{avc})V

_{c}/(f

_{v}Δh

_{vl})][T

_{cb}T

_{cT}/ΔT]

_{v}(s

_{vpi})/dt)] + [ρC

_{p}ln(T

_{avc})V

_{c}]SLR/ΔT = ∆ρ

_{k}f

_{v}V

_{c}

^{3}/2ξ

^{2}SLR

**Figure 4.**The predicted vertical updraft velocity for a saturated cloud developing vertically with the parameters used in Figure 2, namely, V = 5 m/year and dT

_{E}/dt = 0.018 K/year. The Stefan–Boltzmann constant σ = 5.57 × 10

^{−8}W/m

^{2}K

^{4}, emissivity ε

_{α}= 0.85. The current rate of Earth warming is dT

_{E}/dt = 5.7 × 10

^{−10}K/s.

**Figure 5.**The demand for tropospheric entropy generation and entropy generation rates available from clouds are schematically illustrated. The expected clouds and related weather phenomena are also illustrated. Stratus clouds are closer-to-equilibrium clouds (with zero to low entropy production rates). However, they can aid warming. In contrast, cumulus and nimbus clouds are non-equilibrium clouds with an entropy generation rate in the cloud, which increases with the vertical updraft velocity. Multicell development, lighting, and swirls add to the rate.

#### 3.3. Intense Weather and Entropy Generation in Complex Systems

#### 3.4. Analogy with Solidification Patterns (Microstructures)

_{c}/SLR = 2ξ

^{2}[[ξ dΔ(f

_{v}(s

_{vpi})/dt)] + [ρC

_{p}ln(T

_{avc})V

_{c}]SLR/ΔT]/∆ρ

_{k}f

_{v}V

_{c}

^{2}

_{c}/SLR = [(2ξ

^{2}C

_{p}ln(T

_{avc}) ΔT/f

_{v})]

^{0.5}/ΔT

_{interface}= (D/ΔT0)

_{si}T

_{li})/G

_{sli}h

_{sl}] < (V/G

_{s}li) < 2[((dsgen/dt)T

_{si}T

_{li})/G

_{sli}h

_{sl}]

^{2}·f(T

_{av}, Co), where T

_{av}is an average temperature in the two-phase region and Co is the average bulk chemical composition [13,36]. Note also that a wide diffuse-interface condition is noted (like the case of clouds) when the solidification occurs close to the T(C0) temperature, i.e., when the free-energy functions of the liquid and solid intersect in an alloy phase diagram. The T(C0) is the temperature and composition of equal free energy per mole for the phases. A circular repeating reaction with static steady-state patterns like the decaying Belousov–Zhabotinsky (BZ) reaction patterns at small scales is also known as a feature related to the entropy generation rate [69,70,71]. We have not yet identified any similar BZ reactions in clouds.

**Figure 6.**A pictorial analogy of the pattern formations in the diffuse-interface regions of clouds (Column A) and solidification microstructures (with a positive (Column B) and negative (column C) temperature gradient in front of the solidifying interface). Adapted from references [74,75,82,83,84,85].

## 4. Summary, Comments, and Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Description Scheme of Various Phenomena in the Key Equations for Tropospheric and Cloud Entropy Balance

Relevant Equation | Expression | Type | Units |

Equation (5b) | (4A_{Earth}Φε_{χ} σT_{c}^{3}/3)Δt | Entropy lost from the troposphere from radiation with the average cloud temperature | J/m^{3}.K |

Equation (5b) | (4σA_{Earth}(1 − Φ)ε_{α} T_{E}^{3}/3)Δt | Entropy lost from the troposphere from the Earth’s surface that do not see cloud cover | J/m^{3}.K |

Equation (5b) | (4σT_{sun}^{3}/3)κ_{1}ψΔt | Entropy lost from the troposphere from radiation from the Earth that is mitigated by clouds | J/m^{3}.K |

Equation (6) | hρC_{p}(ln(T)) = | Entropy changes per unit area of the tropospheric slice | J/m^{2}.K |

Equation (6) | (4σψT_{sun}^{3}/3)Δt | Entropy changes per unit area of the tropospheric slice | J/m^{2}.K |

Equation (6) | (4σT_{sun}^{3}/3)κψΔt | Entropy exchange not captured in the other terms that arise from reflections | J/m^{2}.K |

Equation (6) | hds_{gen}/dtΔt | Entropy generation term in the troposphere | J/m^{2}.K |

Equation (6) | dΔδ/dt.Δt | Entropy exchange from evaporation and condensation that crosses the control volume boundary located at the Earth’s surface | J/m^{2}.K |

Equation (6) | ΛΔt | Tηε entropy-loss term associated with the catch-all electron jet (like lightning) that crosses the boundaries of the tropospheric control volume | J/m^{2}.K |

Equation (15c) | $SLR=-g(\frac{(1+(\mathsf{\Delta}{S/R\left){r}_{v}\epsilon \right)}^{\text{}}}{(Cp+({\mathsf{\Delta}S}^{2}{/R\left){r}_{v}\mathrm{\u0273}\right)}^{\text{}}})$ | Near-saturation and saturation lapse rate | K/m |

Equation (16) | 2ε_{χ}κ_{1}T_{c(av)}^{3}/ε_{α}.λT_{E}^{3} > 1 | Bifurcation condition | Dimensionless |

Equation (17) | [ξ dΔ(f_{v}(s_{vpi})/dt)]SLR/ΔT | Rate of entropy generation per unit volume with moisture exchange processes | J/m^{3}.K.s |

Equation (17) | [C_{p}ln(T)V_{c(av)}_{c}]SLR/ΔT | Rate of entropy generation with tropospheric warming processes in the vertically expanding cloud | J/m^{3}.K.s |

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**Table 1.**Cloud types and their typical water content and vertical velocity development, adopted from reference [46]. Clouds with “cumulus” in the name show updraft velocities. Stratus clouds develop horizontally. Stratocumulus clouds are hybrids of layered stratus and multicellular cumulus clouds.

Cloud Type | Liquid Water Content (g/m ^{3}) | Measured Vertical Updraft/Velocity (m/s) | Volume Fraction of Water, f_{v} in Clouds |
---|---|---|---|

cirrus | 0.03 | Small | 3 × 10^{−8} |

fog | 0.05 | 0.25 | 5 × 10^{−8} |

stratus | 0.25–0.30 | Small | (2.5–3) × 10^{−7} |

cumulus | 0.25–0.30 | 1 | (2.5–3) × 10^{−7} |

stratocumulus | 0.45 | 0.5 | 4.5 × 10^{−7} |

cumulonimbus | 1.0–3.0 | 10 | (1–3) × 10^{−6} |

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Sekhar, J.A.
An Entropy Generation Rate Model for Tropospheric Behavior That Includes Cloud Evolution. *Entropy* **2023**, *25*, 1625.
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Sekhar JA.
An Entropy Generation Rate Model for Tropospheric Behavior That Includes Cloud Evolution. *Entropy*. 2023; 25(12):1625.
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2023. "An Entropy Generation Rate Model for Tropospheric Behavior That Includes Cloud Evolution" *Entropy* 25, no. 12: 1625.
https://doi.org/10.3390/e25121625