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\begin{document}

\vspace*{2\baselineskip}

\title{On the carbon cycle and its interactions with the biosphere}

\author{Nabil H. Swedan}

\affil{}{Pacific Engineering PLLC, Olympia, Washington, USA}

\abstract{The submitted work reveals that interactions between the carbon cycle and the biosphere may
be calculated using traditional thermodynamic equations. Presently, anthropogenic activities
have changed the natural carbon cycle and climate parameters, which  have direct effects at the
societal and economical levels. For instance, engineering design parameters particularly those
related to civil, environmental, and agricultural projects will not be the same as today. They will
be different as long as the interactions between humans and the environment continue their
present trends. The thermodynamic analysis of these interactions is broad and complex, and the
manuscript is limited to providing parameterizations of the analysis. The observed surface
warming is caused by heat produced by the carbon cycle. Deforestation has averaged nearly
0.19\%  annually since the Industrial\linebreak\newpage}

\noindent\centerline{\parbox{100mm}{Revolution, and its present contribution to surface warming is about 50\%.
The remainder of surface warming is attributed to energy produced by the world, which has a decreasing trend of $-0.82$\%  annually. The contribution of deforestation to surface
warming has a decreasing trend as well, and the net surface greening is increasing by 0.08\%
annually. Approximate current precipitation trend is $-0.31$~mm annually, which is considerably
greater than that of past natural climates. }}


\section{Introduction}

  The carbon cycle involves living matter and inert environment that appear to interact in
accordance with the natural laws of thermodynamics. The green matter belongs to the first level
of the food chain, and non-green living matter depends on the green matter. Variation in the mass
of life in the biosphere thus can be estimated by knowing variation in the content of carbon
dioxide in the atmosphere. Green plants and vegetation convert carbon dioxide and water by
photosynthesis into glucose, cellulose, sugars, carbohydrates and other vegetation basic
constituents. Solar energy is required for photosynthesis, and this process can be thought of as
energy and carbon sink in the carbon cycle, stored in the biosphere as chemical energy. When the
size of life expands, more of the solar energy is converted into chemical energy, and the
surroundings receive less solar heat. The temperature of the surface decreases. Conversely,
plants decay, oxidation, or combustion produce heat of carbon conversion to carbon dioxide that
raises surface temperature. Variation in the size of life thus could have caused past warming and
cooling cycles of the surface. Given that the earth must have reached equilibrium by now,
climate change thermodynamic transformations may be considered infinitesimal displacements
from equilibrium. The equations of thermodynamic equilibrium may thus be extended to the
carbon cycle and traditional thermodynamics used to analyze the interactions between the carbon
cycle and biosphere subsystems.

  Since human invention of farming, the environmental change has disabled the natural
carbon cycle. Energy production adds heat and carbon dioxide to the environment, and these
should be accounted for in the thermodynamic analysis. Deforestation removes heat sink from
the surface, and in turn, solar heat that otherwise would have been stored as chemical energy,
warms the surface. These human related activities may be assessed using proven thermodynamic
relationships that have passed the test of time.

  In this paper, thermodynamic equations related to interactions between biosphere subsystems
are derived, and the significance of the derived equations is then discussed. Sample calculations
are provided so that the calculations may be reproduced by others. The paper discusses results of
the calculations, alignment with observation, and potential use of the work for practical
applications.

\section{Background Information}


  Plants growth follows Liebig's law, or the law of the minimum. The law states that plants
growth is controlled not by the resources available, but by the scarcest resource, or limiting
factor. For the green matter on the surface, water and carbon dioxide are resources for
photosynthesis, and their concentrations are variable with the climate. Consequently, variation in
the size of the green matter may be assessed by knowing carbon dioxide, evaporation, and
precipitation trends. These are climate related parameters, and, therefore, present climate science
understanding and limitations are discussed first. Parameterizations of the thermodynamics of the
carbon cycle is then introduced.

  [\itc{Petit et al.,} \reflink{Peti99}{1999}]         %%% ??? +
  reveals a correlation between carbon dioxide content in the atmosphere
and surface temperature of past climates. If the content of carbon dioxide in the atmosphere
increases, surface temperature increases and vice versa. This publication attributes past climate
changes to variation in carbon dioxide and Milankovitch cycles. The relevant cycle has a
periodicity of $1.00 \times 10^5$~years, and varies the value of the solar constant with time. Because the
present warming trend is three hundred years or less, Milankovitch cycles can be excluded as a
cause of the ongoing climate change.

\begin{sloppypar}
  The current climate understanding attributes surface warming to anthropogenic climate agents
whose net result is solar heat being trapped in the climate system. Carbon dioxide emission is the
major anthropogenic cause. The Intergovernmental Panel on Climate Change (IPCC)
summarizes present climate understanding in reports, the latest of which is
 [\itc{IPCC,} \reflink{IPCC13}{2013}].
Chapter 8 of this report is dedicated to climate drivers through the Radiative Forcing (RF)
concept. The Chapter provides literature for further reading on present-day climate science. RF is
the net change in the energy balance due to imposed perturbations. RF measures variations in the
net radiative flux at the top of the atmosphere, and climate change occurs due to variations in RF.
A relationship between RF and equilibrium global mean surface temperature, $\Delta T$, is derived,
RF$ \;=\lambda$ $\Delta T$. Where $\lambda$ is climate sensitivity (in W m$^{-2}$ \deg K$^{-1}$),
whose value is yet to be determined with certainty.  Derivation of this equation is based on differential
displacement of the climate from radiative equilibrium.
\end{sloppypar}\vspace*{-0.2ex}

  Modern climate models are typically used for investigating the response of climate to various
climate agents. Chapter 9 of
[\itc{IPCC,} \reflink{IPCC13}{2013}]
is dedicated to evaluation and application of climate
models and includes literature for further reading on the subject. An over review of model type is
presented under Section~9.1.2, and a partial list includes: Atmosphere-Ocean General Circulation
Model, Earth System models, Earth System Model of Intermediate Complexity, and Regional
Climate Models. The carbon cycle and other biosphere components appear to be added as new
components to some Earth System Models of climate,
[\itc{IPCC,} \reflink{IPCC13}{2013}]
Chapter 9, Section 9.1.3.1.3,
Section 9.4.5, Tables 9.A.1 and 9.A.2; Chapter 12, Section 12.4.8, Section 12.4.8.2. Climate
models appear to provide no calculated values of deforestation and surface greening, required
biosphere components for this work in order to assess variation in the size of the green matter.
\vspace*{-0.2ex}

  This work reveals that causes of climate variability are internal, within the climate system.
They are not externally forced by the sun. Specifically, interactions of the carbon cycle with the
biosphere caused past variations in climates. Milankovitch cycles are weak and could not have
caused global changes of past climates. The most severe scenario is when orbital eccentricity
varies between 0.05 and 0.005. Variation in the annual average value of the solar constant is
nearly 4.5\%
at the most. This variation is less than its present seasonal variation of approximately 7.5\%.
 At the global level, the maximum solar contribution to past cooling and
warming cycles must, therefore, be less than one degree $\deg$K based on observations
(NOAA, National Oceanic and Atmospheric Administration, National Climate Data Center,
Global Surface Temperature Anomalies, Mean Monthly Surface Temperature Estimates for the
Base Period 1901 to 2000,
\href{https://www.ncdc.noaa.gov/monitoring-references/faq/anomalies.php}{https://www.ncdc.noaa.gov/monitoring-refe rences/faq/anomalies.php}).
% [\itc{NOAA,} \reflink{NOAA00}{2000}].
This is too small compared with the observed past climate surface temperature variations
of 10~$\deg$K to 12~$\deg$K.

  The work presented in this manuscript indicates that heat produced by the carbon cycle warms
the surface, and heat removed by the carbon cycle cools the surface. They can be calculated with
accuracy because they are correlated with variation in the content of carbon dioxide in the
atmosphere and the atmosphere behaves as an ideal gas. In addition, climate change may be
considered as a thermodynamic transformation displaced differentially from equilibrium, which
simplifies the calculations. Surface temperature rise is less than one degree $\deg$K since the
Industrial Revolution, assumed by IPCC as 1750. If compared with surface temperature of 288~$\deg$K,
the entire climate change is in fact a differential displacement from equilibrium. On annual
basis, the observed climate change is an infinitesimal process. Because the earth must have
reached equilibrium prior to the Industrial Revolution, climate change may be assumed as a
thermodynamic process, or a series of them, displaced differentially from equilibrium.

  [\itc{Perry and Green,} \reflink{Perr84}{1984}]
  summarize the current understanding of thermodynamics, whose
laws are primitive and basic laws of nature. It is an established branch of science that addresses
transformations involving energy exchange between thermodynamic systems and their
surroundings. ``A thermodynamic system is set apart mentally from everything else, which is
called the surroundings. The imaginary envelope that separates the system from its surroundings
is the boundary of the system.  A closed thermodynamic system exchanges with its surroundings
energy and matter is not exchanged. The mass of the system remains unchanged. An open system
on the other hand exchanges matter and energy with its surroundings. For open system, the mass
and concentration of its species are variable. When a system at a state of thermodynamic
equilibrium receives heat, it undergoes a spontaneous thermodynamic process, and its
thermodynamic properties vary until a new equilibrium state is reached. During the
thermodynamic process, heat and work are exchanged between system and surroundings.''
 At
equilibrium the total entropy of system and surroundings approaches maximum and Gibbs
Function approaches minimum value.

  The earth as a whole may be considered to exchange negligible matter with outer space. It is
thus a closed thermodynamic system that has been absorbing solar heat for a long time. This is a
spontaneous thermodynamic process and thermodynamic equilibrium must have been reached
long time ago. Neglecting variation in the entropy of outer space based on the third law of
thermodynamics, then the entropy of the earth is at a maximum and Gibbs Function is at a
minimum. They vary with the motion of the earth around the sun, but the variation is
infinitesimal, given that the maximum variation in the solar constant is 3.8\%
in nearly six month.
Likewise, climate change is infinitesimal displacement from equilibrium as discussed earlier. It
is thus reasonable to use the equations of thermodynamic equilibrium to analyze the climate.

  Unlike laboratory experiments or practical applications, the earth's subsystems are large in
size, and the temperature, pressure, and concentration of their species are not always uniform. It
is, however, practical to assume that the atmosphere is well mixed and the concentrations of
atmospheric species are uniform. This fundamental requirement for thermodynamic analysis is
satisfied. The average temperature of the atmosphere may be used, and the external pressure to
which the atmosphere is subjected may be neglected. The reason is that, the atmospheric air
expands and contracts in a surrounding outer space whose pressure is negligible. Any spherical
finite element of the atmosphere has a constant pressure that is equal to the mass of the air above
it. These considerations are important for thermodynamic analysis of the carbon cycle.

\section{Thermodynamics}

  Based on
  [\itc{Perry and Green,} \reflink{Perr84}{1984}],
  the thermodynamic relationships for open or closed
thermodynamic systems with composition change regardless of the cause follow
\begin{equation}\label{1}
d(nG)= -(nS) dT+(nV) dP+ \Sigma \mu_i dn_i                     % (1)
\end{equation}

\begin{equation}\label{2}
\mu_i= [\partial (nH)/\partial n_i]_{nS,\; P,\; n_j}             % (2)
\end{equation}

\begin{equation}\label{3}
[\partial(nG)/\partial T]_P =-nS                              % (3)
\end{equation}
where
$G$ -- molar Gibbs Function of the thermodynamic system, J mol$^{-1}$;

$S$ -- molar entropy of the thermodynamic system, J mol$^{-1}$ \deg K$^{-1}$;

$n$ -- total number of moles in the system;

$n_i$ -- total number of moles of the chemical specie, $i$, present in the system;

$n_j$ -- number of moles of all species other than specie, $i$, to be held as constant;

$T$ temperature of the thermodynamic system, \deg K;

$V$ -- molar volume of the thermodynamic system, m$^3$ mol$^{-1}$;

$P$ -- pressure of the thermodynamic system, Pa;

$\mu_i$ -- chemical potential of the i-th specie, J~mol$^{-1}$;

$H$ -- molar enthalpy of the thermodynamic system, J mol$^{-1}$.

  The carbon cycle process equations in the biosphere and surrounding air follow


\begin{equation}\label{4}
\begin{array}{cccccc}
\mathrm{C}& +& \mathrm{O}_2 &\Rightarrow  \mathrm{CO}_2,&
\Delta \mathrm{H}_c = -393.14      \\                    % (4)      %%% ???
(0)& & (0)  &    (-393.14) &
\end{array}
\end{equation}


\begin{equation}\label{5}
\begin{array}{cccc}
\mathrm{H_2O} (v) &\Rightarrow & \mathrm{H_2O} (l),&	                              \Delta H_v = 43.97  \\                                  % (5)  %%% ???
(-285.57)& &  (-241.60) &
\end{array}
\end{equation}


\begin{eqnarray*}
\begin{array}{lcc}
6\mathrm{CO}_2(g)\; + \;6\mathrm{H_2O} (l)
\stackrel{\mathrm{Light/Chlorophyll}}{\Rightarrow} \\
(-393.14)\;\;\;  (-241.60) \;\;\;  \\
    \\
\Rightarrow \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6(s) + 6\mathrm{O}_2(g) \\
\qquad  (-985)   \quad     \qquad      (0)
\end{array}
\end{eqnarray*}
\begin{equation}\label{6}                                 % (6)
\Delta\mathrm{H}_{c,G} =-2826.0
\end{equation}

Equation \eqref{4}             %  Equation (4)
  occurs spontaneously and produces heat of carbon combustion, oxidation, and
decay of the green matter in the climate system.
Equation \eqref{6}             % Equation (6)
on the other hand is non-spontaneous and requires solar energy to produce glucose, which is the basis for plants and
vegetation constituents. This photosynthesis process removes solar heat from the climate system
and stores it as chemical energy. The values in parenthesis are standard enthalpies of formation
and $\Delta H_c$ and $\Delta H_v$ are the calculated enthalpies of combustion and water evaporation
respectively, all in KJ mol$^{-1}$.
Equation \eqref{5}             % Equation (5)
is presented for the scenario where water vapor
condenses from the atmosphere. This applies to past interglacial periods and the current warming
trend.

\subsection{Atmosphere}

  The atmospheric air is reasonably mixed and homogenous, and the equations of equilibrium
may be used for states differentially displaced from equilibrium. In the proximity of equilibrium,
system entropy is displaced differentially from its equilibrium value and may be assumed as a
constant. As discussed in the background section, the pressure of the atmosphere is constant.
Therefore, the partial differential of
equation \eqref{2}             % Eq. (2)
at constant pressure and entropy conditions may be
eliminated

\begin{equation}\label{7}
\mu_i= \frac{d(nH)}{dn_i}                      % (7)
\end{equation}
  Equation \eqref{7}             % (7)
  reveals that for the specie, $i$, the chemical potential is closely equal to variation
in the enthalpy of the system per one mole of this specie. For carbon dioxide and water vapor,
the chemical potential is thus equal to $\Delta H_c$ and $\Delta H_v$ respectively.

  Theoretically, the atmosphere is an open thermodynamic system and exchanges water vapor
and carbon dioxide with the surface. In practice, variation of these species is so small compared
with the mass of the atmosphere and this mass may be assumed as a constant, a closed
thermodynamic system. In the proximity of equilibrium, $d(nG)=0$, and for a constant
atmospheric pressure, equation
\eqref{3}             % Eq. (3)
yields
$[\partial (nG)/ \partial T]_P =d(nG)/dT=-nS$.
 Equation  \eqref{1}             % (1)
 for the
atmosphere having water vapor and carbon dioxide as variable chemical species gives\pagebreak

\vspace*{-2.1\baselineskip}

\begin{equation}\label{8}
\mu_{\mathrm{H{_2}O}} dn_{\mathrm{H{_2}O}} = - \mu_{\mathrm{CO}_{2}} dn_{\mathrm{CO}_{2}}         %    (8)
\end{equation}

  Substituting the values of $\mu_{\mathrm{H{_2}O}}$ and $\mu_{\mathrm{CO}_{2}}$
  with $\Delta H_v$ and $\Delta H_c$ respectively

\begin{eqnarray*}
dn_{\mathrm{H{_2}O}} = -(\Delta H_c/\Delta H_v) dn_{\mathrm{CO}_{2}} =
\end{eqnarray*}
\begin{equation}\label{9}
(393.14/43.97) dn_{\mathrm{CO}_{2}}
 = 8.94 dn_{\mathrm{CO}_{2}}                          % (9)
\end{equation}

\vspace*{2\baselineskip}

\subsection{Surface of the Earth}

  Referring to equation
  \eqref{6},             % Eq. (6),
  photosynthesis is shown to produce glucose. In reality, the process is
complex, and the products include but not limited to carbohydrates, sugars, cellulous, and others.
Regardless of the nature of photosynthesis products, this process is a heat sink at the surface of
the earth in that solar energy is converted into chemical energy by the green matter and the
surroundings receive less solar heat. Carbon combustion on the other hand,
equation \eqref{4},             %Eq. (4),
 is a heat
supply in the carbon cycle. All of the heat produced by carbon conversion to carbon dioxide is
returned to the surface as latent heat of water vapor condensation,
 equation \eqref{9}.             %Eq. (9).
 For infinitesimal
climate change process, the energy balance of the surface of the earth follows

\begin{equation}\label{10}
d Q_S = dn_{\mathrm{H{_2}O}} \Delta H_v
+ \Delta H_{c,G} dn_G
\end{equation}                               % (10)
 where $Q_S$ is surface heat and $\Delta H_{c,G}$ and $dn_G$ are the heat of combustion and variation in the
number of moles of the green matter respectively, assumed as glucose. Not considered in the
energy balance of the surface,
equation \eqref{10},             %Eq. (10),
radiation terms or internal heat produced in the earth's
core. This is a climate energy balance in the proximity of an equilibrium state. Non-carbon
producing sources of energy cannot be part of
equation \eqref{8}             % Eq. (8)
 or the energy balance,
equation \eqref{10},             % Eq. (10),
   and have
to exit the climate system. Thermodynamics of the carbon cycle does alter radiation rates of the
earth subsystems. However, if the outgoing radiation decreases, the incoming radiation must
decrease equally and vice versa to maintain near equilibrium thermodynamic state.

\section{Significance of the Derived Equations}

  Equation \eqref{9}             % (9)
  reveals that an increase in the content of carbon dioxide in the atmosphere
increases the amount of water vapor in the atmosphere. This implies a decrease in water vapor
condensation and precipitation. To maintain the atmosphere near equilibrium conditions, water
vapor must condense onto the surface. The latent heat of water vapor condensation warms the
surface. The opposite is the case if the content of carbon dioxide decreases with time, and for this
case, more water vapor must condense in the atmosphere instead of the surface. Surface
evaporation increases and surface temperature decreases.

  To calculate present precipitation reduction, a curve fit of carbon dioxide trend with time may
be prepared and
equation \eqref{9}             % Eq. (9)
integrated. Alternatively, the curve may be divided into smaller and
linear segments and precipitation trend calculated using a simple mathematical equation, which
can be derived: For a series of climate thermodynamic processes covering, $n$, years,
equation \eqref{9}             % Eq. (9)
yields

\begin{equation}\label{11}
\Delta \Gamma = -\Delta n_{\mathrm{H_2O}} = - \sum\limits_0^n{_i}
8.94 d(n_{\mathrm{CO_2}})_i
\end{equation}                          %  (11)
where $\Delta \Gamma$ is the total variation in precipitation in, $n$, years, which is equal to the opposite sign
of ($\Delta n_{\mathrm{H{_2}O}}$) and
$d(n_{\mathrm{CO}_{2}})_i$ is variation in the concentration of carbon dioxide in the atmosphere for
the $i$-th year. The concentration of carbon dioxide is typically provided at the end of the year.
Assuming the total change in the number of moles of carbon dioxide at the conclusion of the $n$-th
year is $\Delta n_{\mathrm{CO}_{2}}$, linearity yields
$d(n_{\mathrm{CO}_{2}})_i = (i/n) \Delta n_{\mathrm{CO}_{2}}$.
Equation \eqref{11}             % Equation (11)
gives

\begin{eqnarray*}
\Delta \Gamma = -(1/n) \times  8.94 \times \Delta n_{\mathrm{CO}_{2}}
 {\sum \limits_0^n}{_i}  i =
\end{eqnarray*}
\begin{eqnarray*} %\begin{equation}\label{12}
-(1/n) 8.94 \times \Delta n_{\mathrm{CO_2}}
  (0+n) \times n/2
\end{eqnarray*} %\end{equation}                                   % (12)

\begin{equation}\label{12}
\Delta \Gamma =-4.47 \times n \times \Delta n_{\mathrm{CO_2}}
\end{equation}                                   % (13)

  The content of carbon dioxide in the atmosphere is typically reported in parts per million by
volume, ppmv. The mass of the atmosphere is $5.18 \times  10^{18}$~kg
[\itc{Fleagle and Businger,} \reflink{Flea80}{1980}]
and
the molecular weight of the atmosphere is 28.8.
Equation \eqref{12}             % Equation (13) (12)!
in ppmv carbon dioxide follows

\begin{equation}\label{13}
\Delta \Gamma = -8.04 \times 10^{14} \times n \times
\Delta {\mathrm{ppmv}}_{\mathrm{CO_2}}
\end{equation}                                %    (14) (13)!
where $\Delta {\mathrm{ppmv}}_{\mathrm{CO}_{2}}$ is the total variation in the concentration of carbon dioxide measured in ppmv
at the conclusion of the $n$-th year.

  Referring to
  equations \eqref{9}             % equations (9)
   and
    \eqref{10},             %(10),
     the first term on the right-hand side of
 equation \eqref{10}             % Eq. (10)
      changes
sign with carbon dioxide trend in the atmosphere. If carbon dioxide in the atmosphere has a
decreasing trend, the term is negative and heat is removed from the surface. If, however, the
trend of carbon dioxide is positive, heat is added to the surface.  The second term, $\Delta H_{c,G} dn_G$,
varies sign with the size, or surface cover, of the green matter. If the area covered by plants and
vegetation increases, less solar energy is received by the surroundings and the term is negative.
Conversely, if the size of the green matter decreases, more solar heat is added to the
surroundings and the term is positive. Both of the terms can be calculated theoretically by
knowing trends of carbon dioxide in the atmosphere, and the total energy exchanged with surface
computed. For more clarity,
equation \eqref{10}             % Eq. (10)
will be simplified

\begin{equation}\label{14}
dQ_s = dQ_c + dQ_g
\end{equation}                                       % (15) (14)!

\begin{eqnarray*}
dQ_c = d n_{\mathrm{H_2O}} \Delta H_v =
- \Delta H_c dn_{\mathrm{CO}_{2}} =
\end{eqnarray*}
\begin{equation}\label{15}
393.14
\times 10^3 dn_{\mathrm{CO}_{2}}
\end{equation}                                      %      (16) (15)!

\begin{eqnarray*} %\begin{equation}\label{17}
dQ_g = \Delta H_{c,G} dn_G
\end{eqnarray*} %\end{equation}                                     %       (17)
where
$Q_s$ -- energy of the surface, J;
$Q_c$ -- energy produced by carbon conversion to carbon dioxide, J;
$Q_g$ -- energy produced by the green matter, J.

  Similarly, for linear trend, the procedure used between
  equation \eqref{11}             % equations (11)
   through
    \eqref{13}             %(14) (13)!
   can be
repeated and
equation \eqref{15}             % Eq. (16) (15)!
yields

\begin{equation}\label{16}
\Delta Q_c = 3.54  \times  10^{19} \times n \times \Delta {\mathrm{ppmv}}_{\mathrm{CO}_{2}}
\end{equation}                                   %      (18)  (16)!
 where $\Delta Q_C$ is variation in the heat of carbon conversion to carbon dioxide in, $n$, years.
  Greening of the surface may be estimated from kinetics of the chemical reaction of
  equation \eqref{6}             %Eq. (6)

\begin{equation}\label{17}
K_e = [\mathrm{C}_6\mathrm{H}_{12} \mathrm{O}_6][\mathrm{O}_2]^6 / [\mathrm{H}_2\mathrm{O}]^6[\mathrm{CO}_2]^6
\end{equation}                                  %    (19) (17)!
where $K_e$ is the constant of equilibrium of photosynthesis and the brackets indicate
concentration of products and reactants of the chemical reaction. Given that water and glucose
are virtually in pure liquid and solid phases, their concentrations may be assumed to be equal to
unity. For two consecutive thermodynamic processes having states of equilibrium, a, and, b,
respectively displaced differentially from each other, the constant of equilibrium may be
assumed to be the same.
Equation \eqref{17},             % Equation (19), (17)!
applied twice for states, a, and, b, yields

\begin{eqnarray*}  % \begin{equation}\label{20}
[\mathrm{O}_2]_a^6 / [\mathrm{O}_2]_b^6
= [\mathrm{CO}_2]_a^6 / [\mathrm{CO}_2]_b^6
\end{eqnarray*}  % \end{equation}                                   %       (20)
\begin{eqnarray*}
([\mathrm{O}_2]_a -[\mathrm{O}_2]_b) / [\mathrm{O}_2]_b =
\end{eqnarray*}
\begin{equation}\label{18}
([\mathrm{CO}_2]_a - [\mathrm{CO}_2]_b) / [\mathrm{CO}_2]_b
\end{equation}                                   %     (21) (18)!

  If the concentration is expressed in molar fraction and the states of equilibrium are
infinitesimally apart, then
equation \eqref{18}             % Eq. (21) (18)!
can be simplified

\begin{equation}\label{19}
dy_{\mathrm{O}_2} / y_{\mathrm{O}_2} =
dy_{\mathrm{CO}_2} / y_{\mathrm{CO}_2}
\end{equation}                                  %        (22)(19)!
where $y_i$ is the molar fraction of the $i$-th specie, equals to the number of mole of the specie
divided by the total number of moles. Because oxygen is a product of reaction, the term on the
left hand side of
 equation \eqref{19}             % Eq. (22) (19)!
 is also equal to $dn_G/n_G$ based on
 equation \eqref{6},             %Eq. (6).
 where $n_G$ is the number of
moles of glucose produced. In parts per million by volume of carbon dioxide in the atmosphere,
the percent change in surface greening may thus be obtained
\begin{eqnarray*}
\% \mbox{Greening} = 100 \times dn_G / n_G =
\end{eqnarray*}
\begin{eqnarray*}
100 \times dy_{\mathrm{O}_2}/y_{\mathrm{O}_2} =
\end{eqnarray*}
\begin{equation}\label{20}
 100 \times
 (d{\mathrm{ppmv}}_{\mathrm{CO}_2} /2)/{\mathrm{ppmv}}_{\mathrm{CO}_2}
\end{equation}                                                                     %    (23)(20)!

  The division of $d{\mathrm{ppmv}}_{\mathrm{CO}_{2}}$ by 2 in
  equation \eqref{20}             %Eq. (23)
  is required because variation in the content of
carbon dioxide, $d{\mathrm{ppmv}}_{\mathrm{CO}_{2}}$, is provided at the end of the year. Carbon dioxide concentration,
ppmv$_{\mathrm{CO}_{2}}$, is the average content of carbon dioxide in the atmosphere in the period of time in
consideration. The calculated \%
Greening of the surface by
equation \eqref{20}             % Eq. (23)(20)!
does not include
deforestation. To obtain Net \%
Greening, the percent deforestation must be subtracted from the
value calculated by equation
 \eqref{20}.             %Eq. (23). (20)!

  It should be noted that changes in surface greening includes changes in the mass and size of
plants and vegetation. Variation in the mass of the green matter can be calculated practically
accurately for past climates by knowing variation in the content of carbon dioxide in the
atmosphere. This, however, does not necessarily give variation in the area or surface cover of the
green matter. A vertical growth of forests decreases the content of carbon dioxide in the
atmosphere, but the incident and absorbed solar energy may remain about the same. An increase
in surface cover of forests, or surface greening, on the other hand decreases the rate of solar heat
to the surroundings. Therefore, by calculating energy contribution of the green matter to surface
warming, deforestation can be computed for present climate change.


\section{Sample Calculations}

  Publications
  [\itc{Petit et al.,} \reflink{Peti99}{1999};
  NOAA, National Oceanic and Atmospheric Administration, Earth System Research Laboratory,
Global Monitoring Division, 2018,
\href{https://www.esrl.noaa.gov/gmd/ccgg/trends/data.html}{https:// www.esrl.noaa.gov/gmd/ccgg/trends/data.html}]
  pro\-vi\-de records of present and past
concentration of carbon dioxide in the atmosphere. These can be used to calculate trends of
precipitation. For the period between 2005 and 2010, variation in the concentration of carbon
dioxide $\Delta {\mathrm{ppmv}}_{\mathrm{CO}_{2}}$  was 10.4. Using
 equation \eqref{13},             % Eq. (14),(13)!
 variation in precipitation $\Delta \Gamma =-4.18 \times  10^{16}$ mol
water, nearly $-7.53 \times 10^{14}$~kg water in five years. At surface area of $5.1 \times 10^{14}$~m$^2$, annual
variation in precipitation is $-0.29$~mm annually. Using
equation  \eqref{16}             %Eq. (18) (16)!
the heat produced by carbon
conversion $\Delta Q_c$ is $1.84 \times 10^{21}$~J. The annual heat produced by the carbon cycle $dQ_c$ is
approximately equal to $3.68 \times 10^{20}$~J~yr$^{-1}$.

  The observed heat gained by the surface can be calculated by knowing either sea surface
temperature rise or sea level rise. The total energy gained by the surface of the earth is enthalpy
that raises surface temperature and latent heat that melt ice masses. For differential displacement
from equilibrium, it can be demonstrated that they are equal. Neglecting land for it has a small
thermal capacity, the total heat gained by the surface can thus be calculated by either of the
following equations

\begin{equation}\label{21}
dQ_s = 2  A_r \times A \times d \times Cp_w \times dT_s
\end{equation}                                         %      (24) (21)!

\begin{equation}\label{22}
dQ_s = 2  A_r \times \mathrm{SLR} \times A \times Ls
\end{equation}                                   %          (22) (22)!
\begin{sloppypar}
\noindent
where
$A_r$  -- area ratio between surface water and total surface area of the earth, $\approx 0.7$;
$A $ -- area of the surface of the earth, $\approx 5.1  \times 10^{14}$~m$^2$;
$d$  -- depth of the mixed layer of surface water, $\approx 100$~m;
$Cp_w$  -- specific heat of surface water, \mbox{$\approx 3.90 \times 10^3$ J~kg$^{-1}$ \deg K$^{-1}$;}
$T_s$ -- surface temperature, \deg K;
SLR -- sea level rise, mm;
$L_s$  -- latent heat of ice melting, $3.34 \times 10^5$ J kg$^{-1}$.
\end{sloppypar}

  The used depth of 100 m for the mixed layer is based on
  [\itc{de Boyer Mont\'{e}gut et al.,} \reflink{Boye04}{2004}].
%[\itc{University of Colorado,} \reflink{Univ17}{2017}]        %%% ???
University of Colorado (CU Sea Level Research Group, 2018 rel-1: Global Mean Sea Level
Times Series (Seasonal Signal Removed), 2018,
\href{http://sealevel.colorado.edu/}{http://sealevel. colorado.edu/})
provides sea level rise since 1993. Between 2005 and 2010, sea
level rose by 14.87~mm. Therefore, the energy gained by the surface $\Delta Q_s$ is $3.55 \times 10^{21}$~J based
on
equation \eqref{22},             % Eq. (25), (22)!
 and the annual energy gained $dQ_s =7.09 \times  10^{20}$~J yr$^{-1}$. From
equation \eqref{14},             % Eq. (15), (14)!
 the annual
contribution of the green matter to surface warming $dQ_g$ is $7.09 \times 10^{20}-3.68 \times 10^{20} =3.41 \times 10^{20}$~J.
It can be demonstrated that the ratio $dQ_g/dQ_s$ is equal to deforestation fraction with respect to
equilibrium forest inventory. This ratio is approximately equal to 0.481 of forest inventory prior
to the Industrial Revolution assumed 1750. Therefore, deforestation through 2010 is nearly
48.1\%;
this is about 0.185\%
of average annual deforestation. For the period of time in
consideration, 2005--2010, the average content of carbon dioxide in the atmosphere is 384.74
ppmv$_{\mathrm{CO}_{2}}$ and the average annual variation in the content of carbon dioxide is
$d{\mathrm{ppmv}}_{\mathrm{CO}_{2}}=2.02$.
Therefore, \%
Greening of the surface is 0.263\%
annually,
equation \eqref{20}.             %Eq. (23). (20)!
The Net annual \%
Greening
of the surface is the difference between annual \%
Greening and annual \%
Deforestation,
approximately equal to $0.263\% -0.185\% =0.08\%$.

  Following the last Glacial Maximum during the period of time 18,470--13,800 Before Present
(BP), surface temperature increased by 6\deg K. Using
equation \eqref{21},             %Eq. (24), (21)!
 the surface thus gained
$ 1.67 \times 10^{24}$~J
during that period of time.




  The world energy production can be estimated since 1990 based on energy consumption
records complied by
[\itc{EIA,} \reflink{EIA16}{2016}].
Energy is reported in British Thermal Units, which should be
converted to Joules for comparison with the calculations. For example,
Figure 1--2,
page 8, of
[\itc{EIA,} \reflink{EIA16}{2016}]
reveals that OECD countries consumed in 2011 nearly 240 quadrillion BTU. Non-OECD countries consumed approximately 310 quadrillion BTU in that year. The world energy
production for 2011 is thus equal to ($240+310$)~BTU$ \: \times 1055.06$~J~BTU$^{-1} \times 10^{15}$~Quadrillion$^{-1} =5.80 \times 10^{20}$~J.





\section{Discussion}

\begin{sloppypar}
The thermodynamic processes of the carbon cycles presented in this work will be discussed
for compliance with the laws of thermodynamics.
Equation \eqref{4}             % Equation (4)
is a spontaneous reaction and
does not require external source of energy. Light is required for the photosynthesis reaction of
equation \eqref{6},             %Eq.(6),
which is available in sunrays. Evaporation and condensation of water vapor on the other
hand,
equation \eqref{5},             % Eq. (5),
merit discussion. During cooling cycles, when the concentration of carbon dioxide
in the atmosphere decreases with time, water vapor content in the atmosphere decreases based on
equation \eqref{9}.             % Eq. (9).
This prompts more surface evaporation and surface cooling. Surface water evaporation is
a spontaneous process for the surface is a heat reservoir and its temperature is greater than the
temperature of the atmosphere. When, however, the content of carbon dioxide increases such as
the present time or past interglacial periods, water vapor has to condense on the surface, and
surface temperature increases. This process is equivalent to heat transfer from the cold
atmosphere to the warmer surface, which is a non-spontaneous process and requires external
mechanical work. It is available, and the atmosphere provides such a work in the form of
potential energy. Presently, the height of the atmosphere is decreasing, and this decrease in
potential energy of the atmosphere provides the required work.
[\itc{Thompson and Solomon,} \reflink{Thom02}{2002}]
observed reduction in geopotential heights of the upper atmosphere. The decreasing trends
examined at the South Pole were assumed to be attributed to ozone layer depletion. However,
greenhouse gas emissions are not excluded as a cause. For lack of data relative to trends of
geopotential heights, the observed reduction at the South Pole could be typical elsewhere.
Therefore, all of the thermodynamic processes of the carbon cycle discussed are feasible from a
thermodynamic prospective.
\end{sloppypar}




\begin{sloppypar}
\tabref{1}   and   \tabref{2}        summarize carbon cycle parameters for selected periods of times where past
climate transformations may be assumed linear with time. The period 18,470--13,800 Before
Present (BP) followed the last Glacial Maximum, characterized by severe cold and dry climate
[\itc{Van der Hammen,} \reflink{Hamm74}{1974}].
Cold surface temperature, reduced carbon dioxide content in the
atmosphere, glaciers cover of land, and decreased of water availability; all are unfavorable % to
for life. Widespread biological decay, reduced photosynthesis, and significant increase in solar
energy available to the surroundings occurred. These increased surface temperature and carbon
dioxide content in the atmosphere.
Line 7 of
\tabref{1}           %  Table 1
reveals a negative heat contribution of the
green matter to surface warming, thus suggesting an increase in surface greening by
approximately 0.003\%
annually,
\tabref{2}.           % Table 2.
The total contribution of this warming campaign to
surface greening was nearly 12\%
in approximately $4.7 \times  10^3$~years. This was typical for all past
interglacial periods.
\end{sloppypar}


\begin{sidewaystable}                                   % Table 1
\tablewidth{155mm}
\ftablabel{Summary of Past Climates Energy}
\vspace{5pt}
\footnotesize
\begin{tabular}
{@{}l@{\hspace{5pt}}
l@{\hspace{5pt}}
c@{\hspace{10pt}}
c@{}}
\hline
\\ [-7pt]
Line     & Period             &   18,470--13,800   &  112,320--106,500   \\
         &   Description      &      (BP)          &     (BP)   \\
   [7pt]  \hline   \\ [-7pt]
1  &  Climate transformation duration, years                                    &  4670     &  5820      \\
2  &  Surface temperature change, \deg C                                        &  6.00     &  $-4.50$     \\
3  &  Variation in carbon dioxide, ppmv                                         &  57.00    &  $-35.00$    \\
4  &  Annual precipitation trend, mm                                            &  $-0.002$   &  0.001     \\
5  &  Average annual energy exchanged with the surface, $dQ_s$, J               &  3.58E20  &  $-2.15$E20  \\
6  &  Average annual heat produced by the carbon cycle, $dQ_c$, J               &  2.02E21  &  $-1.02$E21  \\
7  &  Average annual heat added to the surface by                               &  $-1.66$E21 & 1.02E21    \\
   &  the green matter, $dQ_g$, J                                               &           &        \\
[4pt] \hline
\end{tabular}
\end{sidewaystable}


\begin{sidewaystable}                                   % Table 2
\tablewidth{155mm}
\ftablabel{Past Climate Parameters}
\vspace{5pt}
\footnotesize
\begin{tabular}
{@{}l@{\hspace{5pt}}
l@{\hspace{10pt}}
c@{\hspace{10pt}}
c@{}}
\hline
\\ [-7pt]
Line     & Period            &   18,470--13,800   &  112,320--106,500   \\
         &   Description     &      (BP)          & (BP) \\
   [7pt]  \hline   \\ [-7pt]
1  &  Climate transformation duration, years                                &  4670     &  5820      \\
2  &  Average concentration of carbon dioxide,  ppmv                        &  235.00     &  250.00     \\
3  &  Average annual change in carbon dioxide,  ppmv                        &  0.012    &  $-0.006$    \\
4  &  Annual \% Greening of the surface                                     &  0.003   &  $-0.001$     \\
5  &  Annual \% Deforestation of the surface                                &  0.000  &  0.000  \\
6  &  Net annual \% Greening of the surface                                 &  0.003  &  $-0.001$  \\
[4pt] \hline
\end{tabular}
\end{sidewaystable}



  Conversely, the cooling period of time between 112,320--106,500 BP reveals contraction in the
size of the green matter, line 7 of
\tabref{1}.           % Table 1.
Plants contributed heat to surface warming and
simultaneously their mass increased. This is evident from the decreases in the content of carbon
dioxide in the atmosphere, line 3 of
\tabref{1}.           % Table 1.
 Apparently, during severe cooling periods, the
green matter increases in density and vertical growth, not necessarily in size or surface cover.
This is in contrast with warming periods where the growth appears to be predominately
horizontal and sporadic.

\begin{sidewaystable}                                   % Table 3
\tablewidth{155mm}
\ftablabel{Summary of Present Climate Energy}
\vspace{5pt}
\scriptsize
\begin{tabular}
{@{}l@{\hspace{3pt}}
l@{\hspace{3pt}}
c@{\hspace{5pt}}
c@{\hspace{5pt}}
c@{\hspace{5pt}}
c@{}}
\hline
\\ [-7pt]
Line     & Period          &   1993--1998  &  2000--2005   &  2005--2010  &  2010--015 \\
         &   Description   &               &             &               & \\
   [7pt]  \hline   \\ [-7pt]
1  &  Climate transformation duration, years             &  5     &  5      &  5      &  5     \\
2  &  Sea level rise, mm                               &  15.83   &  16.63  &  14.87  &  16.63 \\
3  &  Variation in carbon dioxide, ppmv           &  9.60	& 10.25	& 10.40	& 10.93 \\
4  &  Annual precipitation trend, mm	         & $-0.27$  &  $-0.29$  &  $-0.29$  &  $-0.31$ \\
5  &  Average annual energy produced by the world, J  &  4.10E20  &  4.91E20  &  5.41E20  &  5.92E20 \\
6  &  Average annual energy exchanged with the   &  7.55E20  &  7.93E20  &  7.09E20  &  7.93E20 \\
   &  surface, $dQ_s$, J                         &           &           &           &          \\
7  &  Average annual heat produced by the   &  3.40E20  &  3.63E20  &  3.68E20  &  3.87E20 \\
   &  carbon cycle, $dQ_c$, J                        &           &           &           &          \\
8  &  Average annual heat added by the   &  4.15E20  &  4.30E20  &  3.41E20  &  4.06E20 \\
   &  green matter, $dQ_g$, J                       &           &           &           &          \\
9  &  Annual trend of the heat added to the surface by            &  \multicolumn{4}{c}{$-3.00$E19}   \\
   &  the green matter,     J                                               &           &        \\
[4pt] \hline
\end{tabular}
\end{sidewaystable}

\begin{sidewaystable}                                   % Table 4
\tablewidth{155mm}
\ftablabel{Present Climate Parameters}
\vspace{5pt}
\scriptsize
\begin{tabular}
{@{}l@{\hspace{6pt}}
l@{\hspace{6pt}}
c@{\hspace{6pt}}
c@{\hspace{6pt}}
c@{\hspace{6pt}}
c@{}}
\hline
\\ [-7pt]
Line     & Period          &   1993--1998  &  2000--2005   &  2005--2010  &  2010--2015 \\
         &   Description   &               &             &              & \\
   [7pt]  \hline   \\ [-7pt]
1  &  Climate transformation duration, years          &  5     &  5      &  5      &  5     \\
2  &  Average concentration of carbon dioxide, ppmv  &  361.63  &  374.52  &  384.74  &  395.23 \\
3  &  Average annual change in carbon dioxide, ppmv  &  1.920  &  2.050  &  2.020  &  2.190 \\
4  &  Annual \% Greening of the surface   &  0.265  &  0.274  &  0.263  &  0.277 \\
5  &  Annual \% Deforestation of the surface  &  0.224  &  0.215  &  0.187  &  0.195 \\
6  &  Net annual \% Greening of the surface  &  0.041  &  0.059  &  0.076  &  0.081 \\
[4pt] \hline
\end{tabular}
\end{sidewaystable}



  The natural carbon cycle has been disabled since farming was invented, and deforestation has
become a parameter of the cycle. In
\tabref{3}           % tables 3
and
\tabref{4},           % 4,
climate parameters of the present carbon
cycle are tabulated. A comparison between lines 6 and 8 of
\tabref{3}           % Table 3
reveals that present annual
contribution of deforestation to surface warming is nearly 50\%.
 Assuming major deforestations
began around the Industrial Revolution, 1750, it can be concluded that deforestation trend is
0.19\%
annually. This is in good agreement with
[\itc{FAO,} \reflink{FAO16a}{2016a}, \reflink{FAO16b}{2016b}, \reflink{FAO00}{2000}]
 estimate on the
ground of 0.1\%
to 0.25\%.
 As line 9 of
 \tabref{3}           % Table 3
 shows, the heat added by the green matter has a
decreasing trend, and deforestation is thus decreasing. The calculated Net annual percent
Greening of the surface is 0.08\%,
line 6 of
 \tabref{4}.           % Table 4.
 [\itc{FAO,} \reflink{FAO16a}{2016a}]      %%% ???  % a or b?+
 records reveals surface
greening of 0.03\%
annually and
[\itc{Zaichun Zhu et al.,} \reflink{Zaic16}{2016}]
observed annual surface greening of
0.44\%.
 The calculated Net annual percent Greening falls well within the observed range.

  Also, industrialization and fossil fuel burning have become integral part of the
thermodynamics of the carbon cycle for they produce heat and carbon dioxide. Presently, carbon
dioxide trend in the atmosphere is considerably greater than in past climates, line 3 of
\tabref{2}           % tables 2
and
\tabref{4}.           % 4.
The calculated present precipitation trend is $-0.31$~mm annually, nearly 150 times greater
than past climates, line 4 of
\tabref{1}
and
\tabref{3}.               % tables 1 and 3.
[\itc{Philandras et al.,} \reflink{Phil11}{2011}]
observed $-0.3$ to $-0.7$~mm
annual precipitation trend and
 [\itc{Longobardi and Villani,} \reflink{Long09}{2009}]
  observed $-0.6$~mm annually. The
calculated trend falls within the observed range. A comparison between lines 5 and 7 of
\tabref{3}               % Table 3
reveals that only 65\%
of the present energy produced by the world remains in the climate, a
decrease from 83\%
for the period of time 1993--1998. Approximate annual decrease in the
contribution of energy production to surface warming is nearly $-0.82$\%
 annually. The decrease
may be attributed to using energy sources that produce less net molecules of carbon dioxide than
carbon. This is in line with the present practice of using renewables, biomass, nuclear, and
methane as sources of energy.



\begin{sloppypar}
  The observed glacier melting and disappearance can only be explained by latent heat transfer
of water vapor condensation onto the surface, which support this work. Heat transfer with phase
change of water vapor can provide sufficient heat transfer coefficient to melt glaciers at the
observed ambient temperature rise of 0.01\deg K annually. Other heat transfer phenomena cannot
explain this melting based on practical engineering applications
[\itc{Perry and Green,} \reflink{Perr84}{1984}].
[\itc{Berezovskaya et al.,} \reflink{Bere04}{2004}]
supports this conclusion as well. This publication reveals a decrease
in precipitation in the large Siberian watersheds (Ob, Yenisei, and Lena rivers), yet river runoffs % are
increased. Only by water vapor condensation can this observation be explained. A decrease
in precipitation by 1.0~mm produces one kilogram of water vapor per square meter annually
based on this work. When condensed on permafrost, about 8.4~mm of total melt is produced.
Therefore, river runoffs increased even though precipitation decreased with time.
\end{sloppypar}

\begin{sloppypar}
  The calculations reveal that most of anthropogenic heat production has contributed to surface
warming. Because the world is unevenly populated, surface warming varies at the hemispheric
and regional levels, which is observed
[\itc{IPCC,} \reflink{IPCC13}{2013}].
The majority of the world population is in
the northern hemisphere, and this hemisphere is thus expected to warm more than the southern
hemisphere.
\end{sloppypar}


  As discussed in the background section, climate models do not appear to provide calculated
values of deforestation and surface greening. Therefore, comparison of most of the computed
and relevant climate parameters for the objectives of this work with climate models cannot be
made at this time. The report,
[\itc{IPCC,} \reflink{IPCC13}{2013}],
 reveals that between January 2001 and December
2012 the earth has been steadily accumulating energy at the rate of $0.5 \pm 0.43$~W~m$^{-2}$, Chapter 2,
Section 2.3.2. The calculated energy added to the surface during that period of time is nearly
$8.30 \times 10^{21}$~J, line 6 of
\tabref{3}.               % Table 3.
In Radiative Forcing terms, Wm$^{-2}$, the calculated value of
accumulated energy by climate models is practically equal to that calculated in this work.

\section{Conclusions}
\enlargethispage{-.85\baselineskip}

  The discussion section reveals agreements between calculations and basics of
thermodynamics and observations. It is thus reasonable to conclude that the interactions between
the carbon cycle and biosphere subsystems may be calculated using traditional thermodynamic
equations. The heat produced by the carbon cycle warms the surface of the earth, and the heat
removed by the carbon cycle cools the surface. The procedure presented is not limited to
computing observed changes in climate parameters but also their projected values into the future.
If carbon dioxide emission and deforestation projection scenarios are given,
equation \eqref{16}             %Eq. (18)
and
 equation \eqref{20}             %Eq.(23)
 can be used to calculate $\Delta Q_c$ and \%
 Greening practically accurately. Precipitation trend can
be computed by
equation \eqref{13}.             %Eq. (14).
Equation \eqref{14}             % Equation (15)
 can then be applied to calculate the heat added to the
surface, $dQ_s$, and sea level and sea surface temperature rise obtained by
equations \eqref{21}             % equations (24)
and
 \eqref{22}.             %(25).

  Clearly, understanding the carbon cycle and its thermodynamic interactions with human
activities and other biosphere subsystems is important because the environment is impacted by
these interactions. Uneven distribution of surface warming at the hemispheric and regional levels
must alter global and regional hydrology. This includes but not limited to trends of precipitation,
open water evaporation, and frequency of intense precipitation. Basis of engineering design will,
therefore, vary particularly for civil, agriculture, and environment related projects. For instance,
projecting evaporation and precipitation trends in a warming world are critical for projects such
as dams, retention lagoons, zero discharge ponds, irrigation, and others. Similarly, sea level rise
can have considerable impact on the feasibility of projects located in the proximity of surface
water. Surface deforestation and greening are important for sustainable forest management. This
work may be used to assess the general state of world forests without requiring forest inventories
on the ground, and may assist in projecting basis of engineering design.

\acknowledgments{My sincere thanks to the team of Russian Journal of Earth Sciences for their review and comments. Mrs. Olga Aleksanova and Mr. Stanislav Nekhoroshev are gratefully acknowledged for handling this manuscript. The time and effort of those who contributed to this work directly or indirectly are appreciated.}


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\end{document}







