%&latex
\subsection{The conservation laws of magnetohydrodynamics}
\label{Sec:TheEquations}
%
\subsubsection{Overview}
%
The core of a climate model 
must be a set of equations 
describing the equations of fluid flow, 
namely the Navier-Stokes equations\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Scaife2007,Chorin1993}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The Navier-Stokes equations are nonlinear 
partial differential equations, which, in general,  
are impossible to solve analytically.
In very special cases numerical methods lead to 
useful results, but there is no systematics for 
the general case. In addition, the Navier-Stokes
approach has to be extended to multi-component
problems, which does not simplify the analysis. 

Climate modelers often do not accept that
\lq\lq climate models are too complex and 
uncertain to provide useful projections 
of climate change\rq\rq\TschSpace%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Mitchell2007}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Rather, they claim that 
\lq\lq current models enable [them] to attribute 
the causes of past climate change and predict the 
main features of the future climate with a high 
degree of confidence\rq\rq\TschSpace%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Mitchell2007}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Evidently, this claim 
(not specifying the observables subject to the prediction)
contradicts to what is well-known 
from theoretical meteorology, namely 
that the predictability of the weather forecast 
models is (and must be) rather limited 
(i.e.\ limited to a few days)\TschSpace%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Zdunkowski2003}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The non-solvability of Navier-Stokes-type 
equations is related (but not restricted) 
to the chaotic character of turbulence. 
But this is not the only reason 
why the climate modeling cannot be 
built on a solid ground. 
Equally importantly,
even the full set of equations providing 
a proper model of the atmospheric system
(not to say atmospheric-oceanographic system)
are not known (and never will) to a full extent. 
All models used for \lq\lq simulation\rq\rq\ 
are (and have to be) oversimplified. 
However, in general a set of oversimplified nonlinear 
partial differential equations exhibits 
a totally different behavior than 
a more realistic, more complex 
system. Because there exist no strategy 
for a stepwise refinement 
within the spirit of the renormalization (semi-)group, 
one cannot make any useful predictions.
The real world is too complex to be represented 
properly by a feasable system of equations ready 
for processing\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Zichichi2007}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The only safe statement that can be made is that 
the dynamics of the weather is probably governed 
by a generalized Navier-Stokes-type dynamics. 

Evidently, the electromagnetic interactions
have to be included, leading straightly 
to the discipline of Magnetohydrodynamics (MHD)\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Davidson2001,Gerlich1970,Shu1992a,Shu1992b}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
This may be regarded as a set of equations 
expressing all the essential 
physics of a fluid, gas and/or plasma.

In the following these essential equations are reviewed.
The purpose is twofold:
%
\begin{itemize}
\item Firstly, it should be made a survey 
      of what budget relations really exist
      in the case of atmospheric physical systems.
\item Secondly, the question should be discussed 
      at what point the supposed greenhouse 
      mechanism does enter the equations and where
      the carbon dioxide concentration appears.
\end{itemize}
%       
Unfortunately, the latter aspect seems 
to be obfuscated in the mainstream 
approaches of climatology.

\subsubsection{Electric charge conservation}
%
As usual, electric charge conservation is described
by the continuity equation
%
\begin{equation}
\frac{\partial\varrho_e}{\partial t}
+
\mbox{\boldmath$\nabla$} \cdot \textbf{j}
=
0
\label{Eq:ECC}
\end{equation}
%
where $\varrho_e$ is the electrical (excess) charge density
and $\textbf{j}$ is the electrical (external) current density.
%
\subsubsection{Mass conservation}
%
The conservation of mass is described by another
sort of continuity equation
%
\begin{equation}
\frac{\partial\varrho}{\partial t}
+
\mbox{\boldmath$\nabla$} \cdot ( \varrho\,\textbf{v} )
=
0
\label{Eq:MC}
\end{equation}
%
where $\varrho$ is the mass density
and $\varrho\,\textbf{v}$ is the
density of the mass current.
%
\subsubsection{Maxwell's equations}
%
The electromagnetic fields are described 
by Maxwell's field equations that read
%
\begin{eqnarray}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\mbox{\boldmath$\nabla$} \cdot \textbf{D}
&=&
\varrho_e
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\mbox{\boldmath$\nabla$} \times \textbf{E}
&=&
- \, \frac{\partial\textbf{B}}{\partial t}
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\\
\mbox{\boldmath$\nabla$} \cdot \textbf{B}
&=&
0
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\mbox{\boldmath$\nabla$} \times \textbf{H}
&=&
\textbf{j}
+
\frac{\partial\textbf{D}}{\partial t}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\label{Eq:MXE}
\end{eqnarray}
%
where the standard notation is used.
They have to be supplemented by
the material equations
%
\begin{eqnarray}
\textbf{D}
&=&
\varepsilon\,\varepsilon_0\, 
\textbf{E}
\\
\textbf{B}
&=&
\mu\,\mu_0\, 
\textbf{H}
\label{Eq:MTE}
\end{eqnarray}
%
where $\varepsilon$ and $\mu$ are assumed 
to be constant in space and time, 
an assumption that was already made by Maxwell.

\subsubsection{Ohm's law for moving media}
%
Electric transport is described 
by Ohm's law for moving media
%
\begin{eqnarray}
\textbf{j} - \varrho_e \textbf{v}
&=&
\mbox{\Large\boldmath$\sigma$} 
\, 
( \textbf{E} + \textbf{v} \times \textbf{B} )
\label{Eq:OLS}
\end{eqnarray}
%
with
$\mbox{\Large\boldmath$\sigma$}$ 
being the electrical conductivity tensor. Expressed in terms
of the resistivity tensor
$\mbox{\Large\boldmath$\rho$}$ 
this reads
%
\begin{eqnarray}
\mbox{\Large\boldmath$\rho$}
\, 
( \textbf{j} - \varrho_e \textbf{v} )
&=&
\textbf{E} + \textbf{v} \times \textbf{B}
\label{Eq:OLMM}
\end{eqnarray}
%
\subsubsection{Momentum balance equation}
%
Conservation of momentum is described by
a momentum balance equation, also known 
as Navier-Stokes equation,
%
\begin{equation}
\frac{\partial}{\partial t}
( \varrho\,\textbf{v} )
+
\mbox{\boldmath$\nabla$}
\cdot 
( \varrho\,\textbf{v}\otimes\textbf{v} )
=
- 
\mbox{\boldmath$\nabla$} p
-
\varrho \, \mbox{\boldmath$\nabla$} \Phi
+
\varrho_e \textbf{E}
+
\textbf{j} \times \textbf{B}
+
\mbox{\boldmath$\nabla$} \cdot \textbf{R}
+
\textbf{F}_{\mbox{\scriptsize\rm ext}}
\label{Eq:MBE}
\end{equation}
%
where 
$\textbf{v}$ is the velocity vector field,
$p$ the pressure field,
$\Phi$ the gravitational potential,
$\textbf{R}$ the friction tensor,
and 
$\textbf{F}_{\mbox{\scriptsize\rm ext}}$
are the external force densities, 
which could describe the Coriolis and
centrifugal accelerations.
%
\subsubsection{Total energy balance equation}
%
The conservation of energy is described by
%
\begin{eqnarray}
\frac{\partial}{\partial t}
\left(
\frac{ \varrho }{2}
|\textbf{v}|^2
+
\frac{1}{2} 
\,
\textbf{H} \cdot \textbf{B}
+
\frac{1}{2} \textbf{E} \cdot \textbf{D}
+
\varrho \, \Phi
+
\varrho \, u
\right)
+
\phantom{xxxxxxxxxxxxxxxxxxxx}
&&
\nonumber \\
+
\mbox{\boldmath$\nabla$}
\cdot
\left(
\frac{ \varrho }{2}
|\textbf{v}|^2 \, \textbf{v}
+
\textbf{E} \times \textbf{H}
+
\varrho \, \Phi \, \textbf{v}
+
\varrho \, u \, \textbf{v}
+ 
p\,\textbf{v}
-
\textbf{v} \cdot \textbf{R}
+
\mbox{\large\boldmath$\lambda$}
\cdot 
\mbox{\boldmath$\nabla$} T
\right)
=&&
\nonumber \\
=
\varrho\,
\frac{\partial \Phi}{\partial t}
+
\textbf{F}_{\mbox{\scriptsize\rm ext}}
\cdot
\textbf{v}
+
\textbf{Q}
\,\,
&&
\label{Eq:TEBE}
\end{eqnarray} 
%
where 
$u$ is the density of the internal energy,
$T$ is the temperature field, 
and
$\mbox{\large\boldmath$\lambda$}$
the thermal conductivity tensor,
respectively. Furthermore a term
\textbf{Q} 
has been added which could describe
a heat density source or sink distribution.
%
\subsubsection{Poynting's theorem} 
%
From Maxwell's equation with 
space-time independent $\varepsilon$
and $\mu$ one obtains the relation
%
\begin{equation}
\frac{ \partial }{ \partial t}
\left(
\frac{1}{2}
\,
\textbf{H} \cdot \textbf{B}
+
\frac{1}{2}
\,
\textbf{E} \cdot \textbf{D}
\right)
+
\mbox{\boldmath$\nabla$}
\cdot
( \textbf{E} \times \textbf{H} )
=
- \,
\textbf{j} \cdot \textbf{E}
\label{Eq:PT} 
\end{equation}
%
This relation is a balance equation.
The Pointing vector field 
$\textbf{E} \times \textbf{H}$
may be interpreted as an energy current density 
of the electromagnetic field.%
\subsubsection{Consequences of the conservation laws}
%
Multiplying Ohm's law for moving media 
(Equation \ref{Eq:OLMM})
with 
$( \textbf{j} - \varrho_e \, \textbf{v} )$ 
one gets
%
\begin{eqnarray}
( \textbf{j} - \varrho_e \textbf{v} )
\, 
\mbox{\Large\boldmath$\rho$} 
\, 
( \textbf{j} - \varrho_e \textbf{v} )
&=&
\textbf{j} \cdot \textbf{E} 
+ 
\textbf{j} \cdot ( \textbf{v} \times \textbf{B} )
-
\varrho_e \, \textbf{v} \cdot \textbf{E}
\nonumber\\
&=&
\textbf{j} \cdot \textbf{E} 
- 
\textbf{v} \cdot ( \textbf{j} \times \textbf{B} )
-
\varrho_e \, \textbf{v} \cdot \textbf{E}
\label{Eq:CCL1}
\end{eqnarray}
%
which may be rewritten as
%
\begin{equation}
\textbf{j} \cdot \textbf{E}
= 
( \textbf{j} - \varrho_e \textbf{v} )
\, 
\mbox{\Large\boldmath$\rho$} 
\, 
( \textbf{j} - \varrho_e \textbf{v} )
+
\textbf{v} \cdot ( \textbf{j} \times \textbf{B} )
+
\varrho_e \, \textbf{v} \cdot \textbf{E}
\label{Eq:CCL2}
\end{equation}
%
Inserting this into Poynting's theorem 
(Equation~\ref{Eq:PT}) 
one obtains
%
\begin{eqnarray}
\frac{ \partial }{ \partial t}
\left(
\frac{1}{2}
\,
\textbf{H} \cdot \textbf{B}
+
\frac{1}{2}
\,
\textbf{E} \cdot \textbf{D}
\right)
+
\mbox{\boldmath$\nabla$}
\cdot
( \textbf{E} \times \textbf{H} )
=
\phantom{xxxxxxxxxxxxxxxxxxxx}
&&
\nonumber\\
=
- \,
( \textbf{j} - \varrho_e \textbf{v} )
\, 
\mbox{\Large\boldmath$\rho$}  
\, 
( \textbf{j} - \varrho_e \textbf{v} )
-
\textbf{v} \cdot
(
\varrho_e \, \textbf{E}
+
\textbf{j} \times \textbf{B}
) 
&&
\label{Eq:CCL3}
\end{eqnarray}
%
On the other hand, if one applies
the scalar product with $\textbf{v}$
on the momentum balance equation
(\ref{Eq:MBE}) one gets
%
\begin{eqnarray}
\frac{\partial}{\partial t}
\left( 
\frac{\varrho}{2} \, |\textbf{v}|^2 
\right)
+
\mbox{\boldmath$\nabla$} \cdot 
\left( \frac{\varrho}{2} \, |\textbf{v}|^2 \, \textbf{v} \right)
=
\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}
&&
\nonumber\\
=
- 
\textbf{v} \cdot \mbox{\boldmath$\nabla$} p
-
\varrho \, \textbf{v} \cdot \mbox{\boldmath$\nabla$} \Phi
+
\textbf{v} \cdot
( 
\varrho_e \textbf{E}
+
\textbf{j} \times \textbf{B}
)
+
\textbf{v} \cdot
( 
\mbox{\boldmath$\nabla$} \cdot \textbf{R}
)
+
\textbf{v} \cdot 
\textbf{F}_{\mbox{\scriptsize\rm ext}}
\,\,
&&
\label{Eq:CCL4}
\end{eqnarray}
%
Replacing 
$
\textbf{v} \cdot
( 
\varrho_e \textbf{E}
+
\textbf{j} \times \textbf{B}
)
$
with 
Equation (\ref{Eq:CCL3}) and doing 
some elementary manipulations one 
finally obtains
%
\begin{eqnarray}
\frac{\partial}{\partial t}
\left(
\frac{\varrho}{2}
|\textbf{v}|^2
+
\frac{1}{2}
\,
\textbf{H} \cdot \textbf{B}
+
\frac{1}{2}
\textbf{E} \cdot \textbf{D}
+
\varrho \, \Phi
\right)
+
\phantom{xxxxxxxxxxxxxxxxxxxxxxxxx}
&&
\nonumber \\
+
\mbox{\boldmath$\nabla$}
\cdot
\left(
\frac{\varrho}{2}
|\textbf{v}|^2
\textbf{v}
+
\textbf{E} \times \textbf{H}
-
\textbf{v} \cdot \textbf{R}
+ 
p\,\textbf{v}
+
\varrho\,\Phi\,\textbf{v}
\right)
=
\phantom{xxxxxxxx}
&&
\nonumber\\ 
=
p\,\mbox{\boldmath$\nabla$}\cdot\textbf{v}
+
\varrho\,\frac{\partial\Phi}{\partial t}
-
{\rm Tr} ( (\mbox{\boldmath$\nabla$}\otimes\textbf{v})\cdot\textbf{R})
- \,
( \textbf{j} - \varrho_e \textbf{v} )
\, 
\mbox{\Large\boldmath$\rho$} 
\, 
( \textbf{j} - \varrho_e \textbf{v} )
+
\textbf{F}_{\mbox{\scriptsize\rm ext}}
\cdot 
\textbf{v} 
\,\,
\end{eqnarray}
%
Hence, this relation is a consequence of the 
fundamental equations of magnetohydrodynamics.    
The heat density source term $\textbf{Q}$,
the internal energy density $u$, 
and the divergence of the heat current 
density $\textbf{q}$ are missing here.
%
\subsubsection{General heat equation}
With
%
\begin{equation}
du
=
\frac{p}{\varrho^2}
\,
d\varrho
+
T
\,
ds
\end{equation}
%
for reversible processes one can substitute the density of 
the internal energy $u$ by the density of the entropy $s$.

With the aid of
Equations
(\ref{Eq:TEBE}) 
and 
(\ref{Eq:PT})
one derives a differential equation  
for the entropy density $s$:
%
\begin{eqnarray}
&&
\frac{ \partial(\varrho\,s) }{ \partial t }
+
\mbox{\boldmath$\nabla$} \cdot (\varrho\,s\,\textbf{v})
=
\nonumber \\
&&
\phantom{xxxxxxxxxx}  
=
\frac{1}{T}
\,
{\rm Tr} ( ( \mbox{\boldmath$\nabla$} \otimes \textbf{v} ) \cdot \textbf{R} )
+
\frac{1}{T}
\,
( \textbf{j} - \varrho_e \textbf{v} )
\,
\mbox{\Large\boldmath$\rho$}
\,
( \textbf{j} - \varrho_e \textbf{v} )
\nonumber \\
&& 
\phantom{xxxxxxxxxxxxxxxxxx} 
- \, \frac{1}{T}
\,
\mbox{\boldmath$\nabla$} 
\cdot 
(
\mbox{\large\boldmath$\lambda$}
\cdot 
\mbox{\boldmath$\nabla$} T )
+
\frac{\textbf{Q}}{T}
\label{Eq:GHE}
\end{eqnarray}
%
This is the generalized form 
of the heat equation.

Only with artificial heat densities 
\textbf{Q} 
in Equation (143) one can incorporate a hypothetical
warming by radiation. There is no term that depends on the carbon dioxide concentration.
\subsubsection{Discussion}
%
The equations discussed above 
comprise a system of one-fluid equations only. 
One can (and must) write down many-fluid equations 
and, in addition, the averaged equations describing 
the turbulence.
To get a realistic model of the real world,
the above equations \textit{must} be generalized
to take into account
%
\begin{itemize}
\item the dependency of all relevant coefficients 
      on space and time;
\item the presence and coexistence 
      of various species of fluids and gases;  
\item the inhomogenities of the media,
      the mixture and separation of phases.
\end{itemize}
%
In principle such a generalization will be feasable, if one
cuts the domains of definition  into pieces and treats the 
equations by a method of patches. 
Thus the final degree of complexity may be much 
larger than originally expected arriving at a system 
of thousands of phenomenological equations 
defining non-linear three-dimensional dynamics and heat transfer\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{GerlichSolo1976,GerlichKagermann1977,Emmerich1978}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It cannot be overemphasized that 
even if these equations are simplified
considerably, one cannot determine 
numerical solutions, 
even for small space regions 
and 
even for small time intervals.
This situation will not change 
in the next 1000 years regardless 
of progress made in computer hardware.
Therefore, global climatologists may continue 
to write updated research grant proposals
demanding next-generation supercomputers
\textit{ad infinitum}.
As the extremely simplified one-fluid 
equations are unsolvable, 
the many-fluid equations 
would be more unsolvable,
the equations that include 
the averaged equations describing 
the turbulence 
would be still more unsolvable,
if \lq\lq unsolvable\rq\rq\ had a comparative.



Regardless of the chosen level of complexity,
these equations are supposed to be the backbone
of climate simulations, or, in other words,  
the foundation of models of nature.
But even this is not true: 
In computer simulations heat conduction and friction
are compületely neglected,
since they are mathematically 
described by 
second order partial derivatives
that cannot be represented on
grids with wide meshes. 
Hence, the computer simulations 
of global climatology 
\textbf{are not based on physical laws.}

The same holds for the speculations about the influence 
of carbon dioxide:
%
\begin{itemize}
\item Although the electromagnetic field is included
      in the MHD-type global climatologic equations, 
      there are no terms 
      that correspond to the absorption of electromagnetic
      radiation.
\item It is hard if not impossible to find the point
      in the MHD-type global climatologic equations, 
      where the concentration 
      of carbon dioxide enters the game. 
\item It is impossible to include the 
      radiative transfer equation (\ref{Eq:RTE}) into 
      the MHD-type climatologic equations.
\item Apparently, there is no reference in the literature,
      where the carbon dioxide concentration is implemented
      in the MHD-type climatologic equations.       
\end{itemize}
% 
Hence, one is left with the possibility 
to include a hypothetical warming by radiation by hand  in terms of artificial heat densities $\textbf{Q}$ 
in Equation~(\ref{Eq:GHE}). But this would 
be equivalent to imposing 
the \lq\lq political correctly\rq\rq\ 
requested anthropogenic rise 
of the temperature even from the beginning 
just saving an additional trivial calculation.  

In case of partial differential equations
more than the equations themselves  
the boundary conditions determine 
the solutions. 
There are so 
many different transfer phenomena,
radiative transfer,
heat transfer,
momentum transfer,
mass transfer,
energy transfer,
etc.\
and 
many types of interfaces,
static or moving,
between
solids,
fluids, 
gases,
plasmas, etc.\
for which there does not exist
an applicable theory, such that 
one even cannot write down the
boundary conditions\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Safran1994,Bouali2006}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

In the \lq\lq approximated\rq\rq\ discretized
equations artificial unphysical boundary conditions
are introduced, in order to prevent running the system
into unphysical states. Such a \lq\lq calculation\rq\rq,
which yields an arbitrary result, is no calculation
in the sense of physics, and hence, in the sense of science.
There is no reason to believe
that global climatologists do not 
know these fundamental scientific facts. 
Nevertheless, in their summaries for
policymakers, global climatologists
claim that they can compute the
influence of carbon dioxide on 
the climates.
% 
