%<*theory:ElectronDensityEstimate>
The electron density can be estimated from the initial neutral gas pressure $p_{neutral}$. Assuming that there is only hydrogen in the chamber and that becomes fully ionised upon plasma creation, the number $N$ of neutral gas particles before the discharge and the number of ions (and electrons) within the discharge is the same. The neutral gas follows the ideal gas equation,
\begin{equation}
p_{neutral} = \frac{N}{V_{ch}}k_BT_{room}
\end{equation}
where $V_{ch}$ [m$^3$] is the chamber volume (mind, not the plasma volume) and $T_{room}$ [K] is the room temperature. This allows the calculation of particle number $N$ and, subsequently, also the plasma density $n_e = N/V_p$. The chamber is a torus with a major radius $R=40$ cm and minor radius $a=10$ cm.
%</theory:ElectronDensityEstimate>
%<*theory:safetyFactor>
In the resulting magnetic geometry, field lines are winding helically around a torus surface, which is called the magnetic surface. The tokamak magnetic field consists of such nested magnetic surfaces. The helical structure at each magnetic surface is described by the safety factor ($q$). It gives the number of toroidal turns necessary for the magnetic field line at the given magnetic surface to reach its original position poloidally. On large aspect ratio circular tokamaks (like GOLEM), where the major radius ($R$) is much larger than the minor radius ($r_0$), it can be approximated by:
\begin{equation}
q(r,t)= \dfrac{r}{R} \dfrac{B_t(t)}{B_p(r,t)},
\label{eq:q_def}
\end{equation}
where $R$ is the major radius of the magnetic axis. An illustration for the meaning of the safety factor can be seen on Figure \ref{fig:q}.
\begin{figure}[ht]
\centerline{
\resizebox{80mm}{!}{\rotatebox{0}{\includegraphics{SafetyFactor/q_small.pdf}}}
\resizebox{80mm}{!}{\rotatebox{0}{\includegraphics{SafetyFactor/q_high.pdf}}}
}
\caption{Magnetic field lines in a tokamak for different safety factors.}
\label{fig:q}
\end{figure}
%</theory:safetyFactor>
%<*theory:LawsonCriterion>
The Lawson criterion is a simple threshold for self-sustained thermonuclear fusion plasma burn at optimum temperature $T_e$, and it also includes the energy confinement time $\tau_E$ along with plasma (electron) density $n_e$:
\begin{equation}
n_e \tau_E > 10^{20} \;\rm s m^{-3}.
\label{eq:Lawson}
\end{equation}
More general information on fusion power production and the tokamak concept can be found at the following sites:
\url{http://www.magfuzio.hu}, \url{http://www.iter.org}, \url{http://www.jet.efda.org/}
%</theory:LawsonCriterion>
\GMpart{Ohmic heating power}{PlasmaHeatingPower}
%<*theory:PlasmaHeatingPower>
In the GOLEM tokamak the only heating mechanism is the ohmic heating power $P_{OH}$
resulting from the plasma current $I_{p}$ flowing in a conductor (plasma) with finite resistance $R_{p}$. $P_{OH}$ can be simply calculated as
\begin{equation}
\label{eq:heating-power}
P_{OH}(t) = R_{p}(t) \cdot I_{p}^2(t)
\end{equation}
The explicit time dependence $X(t)$ highlights that the quantities strongly vary
with time.
%</theory:PlasmaHeatingPower>
\GMpart{Central electron temperature}{theory:plasmaResistanceSimple}
%<*theory:plasmaResistanceSimple>
In the quasistationary
phase of the discharge where $\mathrm{d}I_p/\mathrm{d}t\approx 0$ inductive
effects can be neglected and plasma resistance can be simply estimated as $R_p=U_l/I_p$.
%</theory:plasmaResistanceSimple>
\GMpart{Central electron temperature}{theory:CentralElectronTemperature}
\GMpart{Electron density}
In its current state, the GOLEM tokamak does not have any density measurements. However, as electron density is needed for further calculations, we estimate its order of magnitude from the state law of ideal gases.
For the average density it is assumed, that it is constant during the discharge, apart from the dissociation of the hydrogen gas. There is a 30 second delay between the gas filling and the actual shot, which is enough for the gas to reach thermal equilibrium with the chamber wall. Chamber temperature is monitored with respect to the room temperature, and the difference is normally be zero, but should be checked. (If chamber temperature is not measured, room temperature can be used instead.) The ideal gas law is used to give an order of magnitude estimate of the electron density (in particle/$m^3$):
\begin{equation}
n_{avr} = \dfrac{2p_{ch}}{k_B T_{ch}}.
\end{equation}
We have to note that this is a very rough estimate basically for two reasons:
\begin{enumerate}
\item Plasma in the GOLEM tokamak is not fully ionized, which makes us overestimate the electron density.
\item Due to the plasma-wall interaction, 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{enumerate}
The order of magnitude estimate of the average electron density has to be calculated for all discharges with plasma and included in the shot summary table.
\GMpart{Plasma energy}
%<*theory:PlasmaTotalEnergy>
%The 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:
The total thermal energy content is given by the average plasma pressure $p_p$
confined in the plasma volume $V_p$ as $W_p=p_p V_p$. The plasma volume is
approximately $V_{p}\approx 80$ l. The average plasma pressure can be
approximated by the electron pressure calculated from $T_{e0}$ (with the assumed
profile) and the average
electron density $n_e$ under the assumption of ions and electrons being in
thermal equilibrium. The fact that the magnetic field reduces the degrees of
freedom of particles to 2 must be also taken into account. Under these
assumption the formula for $W_p$ reduces to
\begin{equation}
W_{p}(t)=V_{p} \dfrac{ n_{e}(t) k_B T_{e0}(t)}{3}.
\end{equation}
%</theory:PlasmaTotalEnergy>
Uncertainty of this formula is dominated by the uncertainty of our density estimate, which makes it good only for an order of magnitude estimate. Qualitative time trace reflects that of the electron temperature and thus is more reliable.
Nevertheless, plasma energy has to be calculated for all discharges with plasma and the maximum value has to be included in the shot summary table.
%<*theory:powerBalance>
The evolution of the plasma thermal energy $W_{p}$ is governed by the power
balance between the net heating power $P_{H}$ and the loss power $P_{loss}$ as
\begin{equation}
\label{eq:power-balance}
\diff{W_{p}}{t} = P_{H} - P_{loss}
\end{equation}
The loss power $P_{loss}$ characterizes the net energy
losses, primarily in the form of escaped particles and radiation.
%</theory:powerBalance>
\GMpart{Energy confinement time}
%<*theory:tauE>
In the absence of heating power $P_{H}=0$ the plasma energy would
decay due to $P_{loss}$. Under the assumption that the loss mechanisms are the same in the whole plasma it is assumed that $P_{loss}$ is proportional to $W_p$ as
\begin{equation}
\label{eq:Ploss}
P_{loss}=\frac{W_p}{\tau_E}
\end{equation}
where $\tau_E$ is the energy confinement time. After substituting
\eqref{eq:Ploss} into \eqref{eq:power-balance} with $P_H=0$ and solving it is
seen that $\tau_E$ is the characteristic time constant of the exponential decay
of $W_p$ as visualized in~\autoref{fig:tauE}.
In the quasistationary phase of the tokamak discharge, where
$\mathrm{d}W_p / \mathrm{d}t\approx 0$ and using $P_H=P_{OH}$ the confinement time can then be
estimated as
\begin{equation}
\label{eq:tauE}
\tau_E= \frac{W_{p}}{P_{OH}} .
\end{equation}
The dependence of $\tau_E$ on global plasma parameters is currently
unknown and still is the subject of active research. Large databases of experiments from tokamaks all over the
world were analyzed and the result is the scaling law for confinement with ohmic
heating at low density \cite{Goldston84}
\begin{equation}
\label{eq:tauE-L-scaling}
\tau_E\propto n_e B_t^{0.5}I_p^{-0.5}
\end{equation}
%</theory:tauE>
Having an estimate for the plasma energy, the energy confinement time can be estimated. The loss power can be estimated from the energy balance:
\begin{equation}
P_{loss}(t)=P_{OH}(t)-\dfrac{d W_{pl}}{d t}
\end{equation}
We then have to just substitute it into the definition (\ref{eq:tau}) of the energy confinement time:
Given the uncertainty of the input parameters, maximum value for the energy confinement time should be taken with care. Nevertheless, its maximum should be included in the shot summary table.
\subsection{$q=2$ disruptions}\label{sec:disruptions}
When the plasma current grows so strong that the edge safety factor, defined by \eqref{eq:q_def}, reaches the value of 2, a plasma instability resonant to the $q=2$ rational surface destabilizes, and a discharge terminating disruption occurs. This limit of operation is to be attempted to be reached in this task using about \textbf{5} dedicated shots.
We can calculate the poloidal field at the edge (for large aspect ratio circular tokamaks) using Amp\`{e}re's law, as the enclosed current is the total plasma current:
\begin{equation}
B_p(a,t)= \dfrac{\mu_0}{2\pi} \dfrac{I_{pl}(t)}{a},
\end{equation}
where $a$ is the plasma minor radius. Substituting this expression into formula \eqref{eq:q_def}, the safety factor at the edge can be estimated as:
\begin{equation}
\label{eq:q}
q(a,t)= \dfrac{a^2}{R_0} \dfrac{2 B_t(t) \pi}{\mu_0 I_{pl}(t)}.
\end{equation}
Discharges aiming to reach a low $q(a,t)$ need as large plasma current as possible. As we have very limited control over the evolution of plasma current in GOLEM, we can also set the $\tau_{OH}$ time delay to set up a discharge at the declining phase of the toroidal magnetic field, which will constantly decrease the edge safety factor.
In order to monitor the success of our efforts, the evolution of the discharges should be plotted on the Hugill diagram. The Hugill diagram positions a discharge on the plane of two parameters:
\begin{itemize}
\item Inverse edge safety factor: $\dfrac{1}{q(a,t)}$
\item Murakami parameter (normalized density): $\dfrac{n_{avg} R_0}{B_t(t)}$
\end{itemize}
The Hugill diagram serves as an operation envelope for tokamaks. If either the Murakami parameter is too high or the inverse edge safety factor reaches the value of 0.5, the plasma disrupts.
First, the temporal evolution of the dedicated shots aiming $q=2$ disruptions should be plotted on the Hugill diagram. Afterwards, all previous shots could be plotted to check that none reach the region $1/q>0.5$.
