The objective of this section is to derive a correlation between carbon dioxide emitted into the atmosphere and the net heat gained by the earth’s subsystems. The subsystems in consideration are atmosphere, ocean, glaciers, land, and surface biomass. Land is typically neglected (IPCC, 2013), it gains a small amount of heat compared with the total heat exchanged; apparently, because it is a solid thermal insulator and has a small thermal capacity. The geothermal heat flow to the ocean, glaciers, and land originates in the earth’s core, and there appears to be no scientific evidence that its value varies with the content of carbon dioxide in the atmosphere. Consequently, this heat flow is not considered. The theory section is divided into subsections and the relevant equations published elsewhere are briefly explained and summarized in these subsections.
Heat and mass balance
Carnot cycle representation of atmospheric processes has been used for decades (Michaud, 2000; Emanuel, 1991). The medium of heat transfer of the thermodynamic cycle is moist air of the lower atmosphere. Figure 1 is a schematic representation of the atmosphere as an ideal Carnot heat engine cycle where variation in the content of carbon dioxide in the atmosphere is negligible. At average conditions, the earth’s surface water is the heat reservoir at average surface temperature, TS, and the atmosphere is the cold reservoir. Water vapor condenses in the upper troposphere, and at the tropopause the vapor condenses completely. Average temperature of the upper tropopause, TT, may therefore be assumed to be equal to the temperature of the cold reservoir. The medium of heat transfer, moist air, removes surface heat QH by evaporating water. This transformation is represented by isothermal air expansion between points 1 and 2 at surface temperature TS. The air along with water vapor then adiabatically expands in the lower atmosphere from point 2 at the surface to point 3 in the upper troposphere and the work, WA, is produced. This work raises the air mass against gravity and maintains air circulation. Under pressure of the upper atmosphere, water vapor condenses in the upper troposphere. This transformation is represented by isothermal compression from point 3 to point 4, and the heat QC, is rejected to the colder atmosphere. The dry and cold air and condensed water then return to the surface by gravity from point 4 located in the upper troposphere to point 1 at the surface, and the thermodynamic cycle repeats. At average conditions, QH=QC+WA.
Human population consumes energy, which has been mostly fossil fuel based energy, and carbon dioxide is released in the process. Deforestation releases carbon dioxide as well. Surface greening sequester some of the carbon dioxide, and the net amount of carbon dioxide interacts with water vapor in the atmosphere in accordance with equation (8) of Swedan (2019)
dnCO2 µCO2+dnH2O µH2O=0 (1)
Where
dnCO2=Variation in the number of moles of carbon dioxide in the atmosphere, mol.
µCO2 =Chemical potential of carbon dioxide, -393.14 kJ mol-1.
dnH2O=Variation in the number of moles of water vapor in the atmosphere, mol.
µH2O =Chemical potential of water vapor, 43.97 kJ mol-1.
Derivation of Eq. (1) is lengthy and it is available in Swedan (2019). Fundamentally, the source equation is equation 4-125 of Lin et al. (1984), quoted as follows:
“d(nG)=-(nS) dT+(nV) dP+Σ μi dni (2)
μi= [∂(nH)/∂ni] nS, P, nj (3)
[∂(nG)/∂T]P =-nS (4)
Where
G=Molar Gibbs Function of the thermodynamic system, J mol-1.
S=Molar entropy of the thermodynamic system, J mol-1 °K-1.
n=Total number of moles in the system.
ni=Total number of moles of the chemical specie, i, present in the system.
nj=Number of moles of all species other than specie, i, to be held as constant.
T=Temperature of the thermodynamic system, °K.
V=Molar volume of the thermodynamic system, m3 mol-1.
P=Pressure of the thermodynamic system, Pa.
μi=Chemical potential of the i-th specie, J mol-1.
H=Molar enthalpy of the thermodynamic system, J mol-1.”
If the atmosphere is considered as a thermodynamic system, the surrounding outer space exerts a negligible pressure that may be assumed to be constant. Equation (4) gives [∂(nG)/∂T]P=d(nG)/dT=-nS, and d(nG)=-(nS) dT. For the atmospheric air having carbon dioxide and water vapor as chemical species, Eqs. (2), (3) and (4) reduce to Eq. (1). The chemical potentials of carbon dioxide and water vapor are equal to the heat of carbon combustion and latent heat of water evaporation respectively. The chemical potentials of greenhouse gases other than carbon dioxide have been neglected in Eq. (1). They have short life and negligible variation in the number of moles compared with carbon dioxide. For atmospheric air mass of 5.18 x 1018 kg and air molecular weight of 28.8 (Fleagle and Businger, 1980), Eq. (1) yields
dnH2O µH2O=-dnCO2 µCO2=7.07 x 1019 dppmvCO2 (5)
Where
dppmvCO2=Variation in the concentration of carbon dioxide in the atmosphere in parts per million by volume, ppmv.
Referring to Fig. 1, if the atmosphere is considered as a thermodynamic system, at average conditions, the energy balance follows
QH=QC+WA (6)
Where
QH =Heat supply by the heat reservoir, the surface, at average surface temperature, J.
QC =Heat rejected to the cold reservoir, the atmosphere, at average temperature of the upper troposphere, J.
WA =Work produced by the atmosphere, J.
From Eq. (5), if the concentration of carbon dioxide increases in the atmosphere, water vapor forms in the atmosphere. Or, some of surface evaporation remains uncondensed in the atmosphere. The atmosphere thus loses energy that is equal to the latent heat of water vapor dnH2O µH2O. This deviation from average conditions may be represented by the following heat balance
QH=QC-dnH2O µH2O+W’A (7)
Where
W’A=Work produced by the atmosphere after an amount of carbon dioxide that is equal to dnCO2 has been added to the atmosphere, J.
Subtraction of Eq. (6) from Eq. (7) and keeping in mind Eq. (1) gives
dWA=W’A-WA=dnH2O µH2O=-dnCO2 µCO2 (8)
dWA=-dnCO2 µCO2 (9)
Equation (9) expresses equality between variation in the potential energy of the atmosphere and the opposite sign of variation in the chemical potential of carbon dioxide in the atmosphere. If the concentration of carbon dioxide increases, an external work of the force of gravity thus applies onto the atmosphere that is equal to dWA. This work slightly reverses the natural thermodynamic cycle of the atmosphere as shown in Fig. 2. If now all of the earth subsystems (ocean, glaciers, land, green matter, and the surrounding atmosphere) are considered as a thermodynamic “climate” system, the heat exchanged with the climate system includes heat of combustion of fossil fuels, heat of deforestation, and heat removed by green matter. Carbon dioxide is exchanged as well, and simultaneously, water vapor forms in accordance with Eq. (1). Therefore, an amount of heat that is equal to the latent heat of water evaporation dnH2O µH2O is removed from the system. Based on Lin et al. (1984), the first law of thermodynamics for the system in consideration follows
dQF+dQD-dQG-dnH2O µH2O=dU+dW (10)
dU=dH-d (PV) (11)
Where
dQF=Variation in the chemical energy of fossil fuels, J.
dQD=Variation in the chemical energy of deforestation, J.
dQG=Variation in the chemical energy of living green matter, J.
dU =Variation in the internal energy of the climate system, J.
dW =Variation in the mechanical energy or potential energy of the climate system, J.
dH =Variation in the enthalpy of the climate system, J.
P =Pressure applied on the system, Pa.
V =Volume of the system, m3.
The system as defined is surrounded by an empty outer space and P=0. Therefore, Eq. (11) gives dU=dH. Variation in the enthalpy dH is equal to the sum of variation in the heat content of ocean and glaciers, dQS, and variation in the enthalpy of the atmosphere dHA. The heat gained by land is neglected as discussed earlier. Because the work, dW, is exchanged only with the atmosphere as potential energy, dW is equal to the variation in the potential energy of the atmosphere dWA. Also, because the thermodynamic cycle reverses, this work exchanged dWA is not produced by the atmosphere; it is applied on the atmosphere instead by the force of gravity and must have an opposite sign of WA. Therefore, Eqs. (10) and (11) yield
dQF+dQD-dQG-dnH2O µH2O=dQS+dHA-dWA (12)
Where
dQS=Variation in the heat content of ocean and glaciers, J.
dHA=Variation in the enthalpy of the atmosphere, J.
The heat dnH2O µH2O that forms in the atmosphere is latent in nature; it does not add sensible heat to the atmosphere. It adds only an infinitesimal amount of uncondensed water vapor, or gas, having the same temperature as the atmosphere. Therefore, dHA≈0. Because -dnH2O µH2O and -dWA are equal based on Eq. (8), these two terms cancel out and Eq. (12) simplifies
dQS=dQF+dQD-dQG (13)
Equation (13) indicates that variation in the heat content of the ocean and glaciers, dQS, is equal to the variation in the net chemical energy exchanged. This equality may be used to generate the heat and mass balance of climate change for any period of time. Because fossil fuel combustion, deforestation, and surface greening exchange carbon dioxide with the atmosphere, the heat dQS may be obtained by knowing the net variation in the content of carbon dioxide in the atmosphere as well. As discussed earlier, when carbon dioxide is released, the natural heat engine cycle slightly reverses as shown in Fig. 2. The work exchanged, dWA, is equal to the variation in the potential energy of the atmosphere that is equal to 7.07 x 1019 dppmvCO2 based on equations (5) and (9). Therefore
dQS=dQF+dQD-dQG=dWA/ηA=7.07 x 1019 dppmvCO2/ηA (14)
Where
ηA=Efficiency of the atmosphere considered as a Carnot heat engine cycle, dimensionless. The value of ηA is nearly equal to 0.17, estimated in the calculation section.
Surface temperature and sea level rise
The calculated net heat returned to the surface, dQS, may be obtained by either Eq. (13) or Eq. (14). This heat is gained by the surface infinitesimally with time. Surface temperature rises and glaciers melt, and these two processes may be assumed to be in equilibrium. In the calculation method section, it is demonstrated that the heat dQS divides about equally between ocean and glaciers. Therefore, an amount of heat that is equal to dQS/2 raises surface temperature as follows:
dQs/2=MA CPA dTS (15)
E =MA WS (16)
dTS =(dQS/2) x WS/(Γ x CPA) (17)
Where
MA =Mass of the circulated surface dry air, kg yr-1.
CPA =Specific heat of air, 1 000 J kg-1 °C-1.
dTS =Sea surface temperature rise, which is equal to sea air temperature rise, °C.
WS =Air humidity at saturation with sea water, kg water per kg dry air, dimensionless.
E =Annual evaporation, kg yr-1.
Г =Annual precipitation, which is equal to annual evaporation, 4.86 x 1017 kg yr-1.
Equation (17) is obtained by eliminating MA from equations (15) and (16), and the derived Eq. (17) is identical to the published Eq. (4) of Swedan (2020b). The calculated dTS by Eq. (17) is equal to sea air temperature rise. This air is saturated with sea water, it is thus on the saturation curve of the psychrometric chart. Therefore, dTS represents the increase in global air wet bulb temperature. The increase in land air temperature may be estimated
dTL=dWS LV/CPA (18)
Where
dTL=Land surface air temperature rise, °C.
LV=Latent heat of water evaporation at surface conditions, 2 461.3 kJ kg-1.
The calculated rise in the land air temperature by Eq. (18) does not account for the contribution of sea air temperature rise. The average rise in the land surface air temperature is nearly equal to the weighted average value of dTS and dTL. Sea level rise due to glaciers melting may be obtained by dividing the volume of the melted glaciers by sea area
dh=(dQS/2)/(LF x 0.7 x 5.1 x 1014) (19)
Where
dh=Sea level rise, mm.
LF=Latent heat of ice melting, 334 kJ kg-1.
The values 0.7 and 5.1 x 1014 in the denominator of Eq. (19) are the area ratio between surface water and the total surface area of the earth, dimensionless, and the total surface area of the earth, m2.
Chemical energy of fossil fuels, dQF
The chemical energy of fossil fuels is nearly equal to the majority of the cumulative energy production by the world population. The United States Energy Information Administration EIA (2020) has published a series of reports relative to international energy consumption titled “International Energy Outlook.” For example, Figure 1-2 on page 8 of International Energy Outlook 2016 presents annual energy consumption of OECD countries and non-OECD countries. The total energy consumed is equal to the sum of the two. In 1990 and 2010, the energy consumed by the world in those years was 3.70 x 1020 J and 5.68 x 1020 J respectively. International Energy Outlook 2020 gives energy consumption of 6.3 x 1020 J for 2019. Before 1990, annual energy consumption data are unavailable and linearity is assumed. By knowing the annual energy production, the cumulative chemical energy of fossil fuels may be calculated.
Chemical energy of deforestation, dQD
The value of the heat of deforestation dQD may be computed by using the published Eq. (11) of Swedan (2020a), which is quoted below
“QD=-2.22 x 1022 d [(1/(ηmax) e{ηmax (n)}- η n2/2-1/(ηmax)], Where 2.22 x 1022 is equal to the initial biomass inventory, J; d, annual deforestation fraction, dimensionless; ηmax, maximum value of seasonal efficiency of photosynthesis, dimensionless; η, average value of seasonal efficiency of photosynthesis, dimensionless; n, number of deforestation years “
It should be noted that the value of the heat of deforestation calculated by this equation is based on an initial terrestrial biomass of 2.22 x 1022 J for the Industrial Era. To account for earlier deforestation since farming was invented, the initial biomass has been increased to 2.41 x 1022 J as discussed in the data section. The general form of the heat of deforestation is
dQD=-QG0 x d [(1/(ηmax) e{ηmax (n)}-η n2/2-1/(ηmax)] (20)
Where
dQD =Chemical energy of deforestation, J.
QG0 =Initial terrestrial biomass inventory, J.
d =Annual deforestation fraction, dimensionless.
η =Average value of the seasonal efficiency of photosynthesis, dimensionless.
ηmax =Maximum value of the seasonal efficiency of photosynthesis, dimensionless.
n =Number of deforestation years, yr.
Chemical energy of surface greening, dQG
Surface greening is equal to the trend in the efficiency of seasonal photosynthesis with time. By knowing the initial biomass inventory, the heat removed from the surface may be calculated. In analogy with Eq. (17) of Swedan (2020b), the chemical energy of surface greening may be calculated as follows:
QG =QG0 EXP [ηmax (t-t0)] (21)
QG =QG0 EXP [η (t-t0)] (22)
dQG=QG0 EXP [η (t-t0)] x dη x (t-t0) (23)
Where
QG =Biomass inventory at a time t, J.
QG0 =Initial biomass inventory at an initial time t0, J.
dQG =Chemical energy of surface greening, J.
t-t0 =Period of time in consideration, yr.
dη =Annual increase in photosynthesis efficiency, which is equal to annual surface greening fraction, dimensionless.
Equation (21), which is the published Eq. (17) of Swedan (2020b), has been used to calculate seasonal growth of a single tree and field crop. The concentration of carbon dioxide in the atmosphere varies seasonally; however, at the end of the year, the concentration of carbon dioxide does not change because seasonal decay cancels out seasonal growth of green matter. Surface greening is negligible, regardless of the time factor (t-t0). The scenario is different for the present carbon cycle because the concentration of carbon dioxide in the atmosphere increases every year. The annual average value of photosynthesis efficiency η is used instead of its seasonal maximum value ηmax as shown in Eq. (22). Differentiation of this equation yields Eq. (23). The variable in Eq. (23) is the efficiency of photosynthesis η, it is increasing because the concentration of carbon dioxide is increasing. The value of the annual surface greening fraction dη may be obtained from the published Eq. (13) of Swedan (2020b), which is quoted as follows:
“% Greening=100 x (dppmv/2)/ppmv, where dppmv is annual increase in the concentration of carbon dioxide in the atmosphere in parts per million by volume, ppmv; ppmv, is average concentration of carbon dioxide in the atmosphere, ppmv.”
Therefore, the annual surface greening fraction dη for Eq. (23) is equal to
dη=(dppmvCO2/2)/ppmvCO2 (24)
Where
dppmvCO2=Annual increase in the concentration of carbon dioxide in the atmosphere, ppmv.
ppmvCO2 =Average concentration of carbon dioxide in the atmosphere, ppmv.