\def\CentralElectronTemperatureI{\begin{frame} \frametitle{Central electron temperature estimation I \cite{BrotancePhD}}
Specific resistivity of a fully ionized plasma only depends on:
\begin{itemize}
\item electron temperature ($T_e$)
\item effective charge number ($Z_{eff}$)
\end{itemize}
This dependence is quantified by the Spitzer formula \cite{nrl_formulary} and the \textbf{effective charge number} is assumed as $Z_{eff}$ = 2.5.\\[5 pt]
Making estimation of the electron temperature from integrated value of resistivity ($R_{pl}(t)$) is ambiguous, because center of the plasma has
\begin{itemize}
\item Higher temperature
\item Lower resistivity
\item Higher current density
\end{itemize}
\end{frame}}
\def\CentralElectronTemperatureII{\begin{frame} \frametitle{Central electron temperature estimation II \cite{BrotancePhD}}
\begin{minipage}[]{0.49\tw}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/temp_profile.pdf}
\label{fig:temp_profile}
\end{figure}
\end{minipage}
\begin{minipage}[]{0.49\tw}
However, if we use an equilibrium temperature profile \eqref{eq:temp_prof} (Figure \ref{fig:temp_profile}), measured in more detailed measurements \cite{BrotancePhD}, we can estimate one parameter of the profile, which is in this case the \textbf{central electron temperature} ($T_{e0}(t)$):
\begin{equation}
T_{e}(r,t)=T_{e0}(t)\left( 1-\dfrac{r^2}{a^2} \right)^2
\label{eq:temp_prof}
\end{equation}
\end{minipage}
\end{frame}}
\def\CentralElectronTemperatureIII{\begin{frame} \frametitle{Central electron temperature estimation III \cite{BrotancePhD}}
The central electron temperature ($T_{e0}$) is then calculated using Spitzer's resistivity formula. The current density of plasma is
\begin{equation}
j=E\cdot \sigma
\end{equation}
where $\sigma$ is the specific conductivity of plasma given by
\begin{equation}
\sigma (r)=1.544\cdot10^3\cdot\frac{T_e(r,t)^{3/2}}{Z_{eff}},~~~~~~~ [\Omega^{-1}{\rm m}^{-1}, {\rm eV}]
\label{vodivost}
\end{equation}
and the electric field $E$ is assumed constant in the poloidal cross-section:
\begin{equation}
E=\frac{U_{loop}}{2\pi R}.
\label{E-field}
\end{equation}
Plasma current is obtained by integrating current density over the plasma column:
\begin{equation}
I_{pl}=\int\limits_0^a E\cdot\sigma(r)2\pi r dr.
\label{pl-current}
\end{equation}
\end{frame}}
\def\CentralElectronTemperatureIV{\begin{frame} \frametitle{Central electron temperature estimation IV \cite{BrotancePhD}}
Substituting \eqref{vodivost} and \eqref{E-field} in \eqref{pl-current} gives us the formula for the central electron temperature
\begin{equation}
T_e(0)=\left(\frac{R}{a^2_{}}\frac{8\cdot Z_{eff}}{1.544\cdot10^3}\right)^{2/3}\cdot\left(\frac{I_{pl}}{U_{loop}}\right)^{2/3}.
\end{equation}
For the CASTOR/GOLEM tokamak geometry with $a=78$~mm :
\begin{equation}
T_e(0)=89.8\cdot\left(\frac{I_{pl}~[kA]}{U_{loop}}\right)^{2/3}\approx 230~eV.
\end{equation}
\end{frame}}
\def\PlasmaHeatingPower{\begin{frame} \frametitle{Plasma heating power}
\begin{itemize}
\item In the GOLEM tokamak the only heating mechanism of the plasma is \textbf{ohmic heating}.
\item This is resulting from current flowing in a conductor with finite resistivity.
\item The ohmic heating power can be calculated as:
\end{itemize}
\begin{equation}
P_{OH}(t) = R_{pl}(t) \cdot I_{pl}^2(t)
\label{eq:plasma_heating_power}
\end{equation}
where $R_{pl}$ is the resistance of the plasma and $I_{pl}$ is the current flowing in the plasma.
\end{frame}}
\def\ElectronDensity{\begin{frame} \frametitle{Electron density}
The ideal gas law is used to give an order of magnitude estimate of the \textbf{electron density} (in particle/$m^3$):
\begin{equation}
n_{avr} = \dfrac{2p_{ch}}{k_B T_{ch}}.
\label{eq:electron_density}
\end{equation}
where $p_{ch}$ is the pressure of the chamber and $T_{ch}$ is the chamber temperature, which is normally corresponding with the room temperatare.\\[5 pt]
This is a very rough estimate basically for two reasons:
\begin{itemize}
\item Plasma is not fully ionized, which makes us overestimate the electron density.
\item Adsorbed gases are released from the surface of plasma facing components during the discharge. These atoms enter the plasma and can be ionized, thus making us underestimate the electron density.
\end{itemize}
\end{frame}}
\def\PlasmaEnergy{\begin{frame} \frametitle{Plasma energy}
The \textbf{total energy} content can be simply calculated from the temperature, density and volume ($V$), based on the ideal gas law, taking into account the assumed \eqref{eq:temp_prof} temperature profile:
\begin{equation}
W_{pl}(t)=V \dfrac{ n_{avr} k_B T_{e0}(t)}{3}.
\label{eq:plasma_energy}
\end{equation}
The information that the magnetic field reduces the degrees of freedom of the particles to two has been used to derive this formula. Uncertainty of this formula is dominated by the uncertainty of our density estimate \eqref{eq:electron_density}, which makes it good for only an order of magnitude estimate.
\end{frame}}
\def\EnergyConfinementTime{\begin{frame} \frametitle{Energy confinement time}
An important concept regarding the energy balance of the tokamak fusion reactor is the \textbf{energy confinement time} ($\tau_E$). It is the characteristic time of energy loss:
\begin{equation}
P_{loss}= \dfrac{W_{pl}}{\tau_E},
\label{eq:tau}
\end{equation}
where $P_{loss}$ is the power lost and $W_{pl}$ is the total plasma energy.\\[5 pt]
Having an estimate for the plasma energy \eqref{eq:plasma_energy}, the energy confinement time can be estimated at the point where the plasma energy has its maximum:
\begin{equation}
\tau_E(t_{top})=\frac{W_{pl}(t_{top})}{P_{OH}(t_{top})}.
\label{energy_conf_time1}
\end{equation}
\begin{equation}
\dfrac{d W_{pl}}{d t}(t_{top})=0.
\label{energy_conf_time2}
\end{equation}
\end{frame}}
\def\SafetyFactor{\begin{frame} \frametitle{Safety factor}
\begin{itemize}
\item The tokamak magnetic field consists of nested magnetic surfaces.
\item \textbf{Safety factor} ($q$) gives the number of toroidal turns necessary for the magnetic field line at the given magnetic surface to reach its original position poloidally.
\item On large aspect ratio circular tokamaks (like GOLEM), it can be approximated by:
\end{itemize}
\begin{equation}
q(r,t)= \dfrac{r}{R} \dfrac{B_t(t)}{B_p(r,t)},
\label{eq:q_def}
\end{equation}
\begin{itemize}
\item where $R$ is the major radius, ($r_0$) is the minor radius, $B_t(t)$ is the toridal and $B_p(r,t)$ is the poloidal magnetic field.% An illustration for the meaning of the safety factor can be seen on Figure \ref{fig:q}.
\end{itemize}
\end{frame}}
\def\SafetyFactorIllustrationI{\begin{frame} \frametitle{Safety factor - Illustration I}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/safety_factor.pdf}
\label{fig:safety_factor}
\caption{Magnetic field lines in a tokamak for different safety factors.}
\end{figure}
\end{frame}}
\def\SafetyFactorIllustrationII{\begin{frame} \frametitle{Safety factor - Illustration II}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{figs/safety_factor_result.pdf}
\label{fig:safety_factor}
\caption{The time evolution of the safety factor on plasma edge.}
\end{figure}
\end{frame}}
