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\subsection{The laws of thermodynamics}
\label{Sec:Thermodynamics}
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\subsubsection{Introduction}
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At the time of Fourier's publication\TschSpace%
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\cite{Fourier1824a,Fourier1824b}
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the two fundamental laws of 
classical thermodynamics were 
not known.
For each law 
two equivalent versions as 
formulated by Rudolf Clausius
(January 2, 1822 - August 24, 1888),
the founder of axiomatic thermodynamics,
are given by\TschSpace%
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\cite{Clausius1887a,Clausius1887b}:
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\begin{itemize}
\item 
\textbf{\textit{First law of thermodynamics:}}
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\begin{itemize}
\item
\textit{%
In all cases, when work is transformed 
into heat, an amount of heat in proportion 
to the produced work is used up, and vice versa, 
the same amount of heat can be produced by the 
consumption of an equal amount of work.}
\item
\textit{%
Work can be transformed into heat and vice versa, 
where the amount of one is in proportion to 
the amount of the other.}
\end{itemize}
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This is a definition of the
\textit{mechanical heat equivalent}.
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\end{itemize}
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\begin{itemize}
\item
\textbf{\textit{Second law of thermodynamics:}}
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\begin{itemize}
\item
\textit{%
Heat cannot move itself from 
a cooler body into a warmer one.}
\item
\textit{%
A heat transfer from a cooler body into a warmer 
one cannot happen without compensation.}
\end{itemize}
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A fictitious heat engine which works 
in this way is called a 
\textit{perpetuum mobile of the second kind}.
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\end{itemize}
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Clausius examines thoroughly, that the second law 
is relevant for radiation as well, even if 
image formations with mirrors and lenses are taken 
into account\TschSpace%
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\cite{Clausius1887a,Clausius1887b}.
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\subsubsection{Diagrams}
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It is quite useful to clarify 
the second law of thermodynamics 
with (self-explaining) diagrams.
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\begin{itemize}
\item
A steam engine works
transforming heat 
into mechanical energy,
whereby heat is transferred
from the warmth to the cold
(see\ Figure~\ref{Fig:SteamEngine}).
\end{itemize}
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\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_machine_steam_engine_.eps,width=2.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_machine_steam_engine_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_machine_steam_engine_.png}}
}{}
\vspace*{8pt}
\caption{A steam engine works
         transforming heat 
         into mechanical energy.}
\label{Fig:SteamEngine}         
\end{figure}
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\begin{itemize}
\item
A heat pump (e.g.\ a refrigerator)
works, because an external work is 
applied,
whereby heat is transferred
from the the cold to the warmth
(see\ Figure~\ref{Fig:HeatPump}).
\end{itemize}
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\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_machine_heat_pump_.eps,width=2.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_machine_heat_pump_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_machine_heat_pump_.png}}
}{}
\vspace*{8pt}
\caption{A heat pump (e.g.\ a refrigerator)
         works, because an external work is 
         applied.}
\label{Fig:HeatPump}  
\end{figure}
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\begin{itemize}
\item
In a perpetuum mobile of the second
kind heat is transferred from the 
cold to the warmth without external 
work applied 
(see\ Figure~\ref{Fig:PerpetuumMobile2}).
\end{itemize}
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\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_machine_pm2_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_machine_pm2_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_machine_pm2_.png}}
}{}
\vspace*{8pt}
\caption{Any machine which transfers heat 
         from a low temperature reservoir 
         to a high temperature reservoir
         without external work applied
         cannot exist:
         A \textit{perpetuum mobile of the second kind}
         is impossible.}
\label{Fig:PerpetuumMobile2}          
\end{figure}
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\subsubsection{A paradox}
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The use of
\textit{a perpetuum mobile of the second kind} 
can be found in many modern pseudo-explanations 
of the ${\rm CO}_2$-greenhouse effect. 
Even prominent physicists have relied 
on this argumentation. 
One example was the hypothesis of Stichel
already discussed in Section~\ref{Sec:Stichel1995}\TschSpace%
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\cite{Stichel1995}.
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\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{
\centerline{\psfig{file=PictureLarge_machine_pm2_climate_model_.eps,width=2.50in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{
\centerline{\includegraphics[scale=1.00]{PictureSmall_machine_pm2_climate_model_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{
\centerline{\includegraphics[scale=1.00]{PictureLarge_machine_pm2_climate_model_.png}}
}{}
\vspace*{8pt}
\caption{A machine which transfers heat 
         from a low temperature reservoir (e.g.\ stratosphere) 
         to a high temperature reservoir (e.g.\ atmosphere)
         without external work applied,
         cannot exist - even if it is radiatively coupled to 
         an environment, to which it is radiatively balanced.
         A modern climate model is supposed to be 
         such a variant of a perpetuum mobile of the second kind.}
\end{figure}
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The renowned German climatologist 
Rahmstorf has claimed that greenhouse 
effect does not contradict to the 
the second law of thermodynamics\TschSpace%
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\cite{Rahmstorf2007}:
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\begin{quote}
\lq\lq
Some \lq sceptics\rq\ state that the greenhouse effect 
cannot work since (according to the second law of 
thermodynamics) no radiative energy can be transferred 
from a colder body (the atmosphere) to a warmer one 
(the surface). However, the second law is not violated 
by the greenhouse effect, of course, since, during 
the radiative exchange, in both directions the net 
energy flows from the warmth to the cold.%
\rq\rq\
\end{quote}
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Rahmstorf's reference to the second law of thermodynamics 
is plainly wrong. The second law is a statement about heat, 
not about energy.  
Furthermore the author introduces an obscure notion of 
\lq\lq net energy flow\rq\rq. The relevant quantity 
is the \lq\lq net heat flow\rq\rq, which, of course, 
is the sum of the upward and the downward heat flow
within a fixed system, here the atmospheric system.
It is inadmissible to apply the second law for 
the upward and downward heat separately 
redefining the thermodynamic system 
on the fly.

A similar confusion is currently 
seen in the German version of Wikipedia\TschSpace%
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\cite{Wikipedia2007}:
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\begin{quote}
\lq\lq 
Some have problems with the energy that is radiated 
by the greenhouse gases towards the surface of the 
Earth ($150\,{\rm W}/{\rm m}^2$ - as shown above) 
because this \mbox{energy} flows from a colder body 
(approx.\ $-40\,^\circ{\rm C}$) 
to a warmer one 
(Earth's ground approx.\ $+15\,^\circ{\rm C}$) 
apparently violating the second law of thermodynamics. 
This is a wrong interpretation, since it ignores the radiation 
of the Sun (even 6000 K). With respect to the total 
balance the second law is obeyed indeed.%
\rq\rq
\end{quote}
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Obviously, the authors are confusing energy with heat. 
Furthermore, the system in question here is the 
atmospheric system of the Earth including the Earth's 
ground. Since this system is 
\textbf{assumed to be in radiative balance} 
with its environment, and any other forms of 
energy and mass exchange with its environment are 
strictly prohibited, it defines a system 
in the sense of thermodynamics for which the second 
law holds strictly.

The difference between heat, energy 
and work is crucial for the understanding 
of thermodynamics. The second law is a
statement about this difference.
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\subsubsection{Possible resolution of the paradox}
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It may be due to the following approximation 
that something is possible in climate models, 
which contradicts the second law of thermodynamics. 
In the field theoretical description of 
irreversible thermodynamics, the second 
law is found in the statement, that the heat 
flow density and the gradient of the temperature
point into opposite directions 
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\begin{equation}
\textbf{q} = 
-
\mbox{\large\boldmath$\lambda$}
\cdot 
\textbf{grad} \,T
\end{equation}
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In this formula, the heat conduction   
necessarily is a positive definite
tensor. 
In climate models it is customary
to neglect the thermal conductivity
of the atmosphere, which means 
to set it to zero\TschSpace%
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\cite{Hansen1983}.
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\begin{equation}
\mbox{\large\boldmath$\lambda$}
= 
\mbox{\large\boldmath$0$}
\end{equation}
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This could explain, why the numerical 
simulations could produce small effects 
in contradiction to the second law 
of thermodynamics.
To set the heat conduction to zero 
would not be a real violation 
of the second law of thermodynamics as it corresponds to an
approximation of an ideal system:
In spite of the temperature differences 
no heat flow could move from a warmer area 
to a colder one. It would be in accordance 
to the second law, if there were 
no temperature rise.
In the past, the \lq\lq predictions\rq\rq\ 
of the climate models were pointing 
sometimes in this direction, 
as was shown in detail in 
Section~\ref{Sec:ModernWorks}.
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