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\title{Hands-on project : Experiment on GOLEM}
\author{GUIRLET Remy 130028}
\date{2017-11-14}
\begin{document}
\clearpage\setcounter{page}{1}\pagestyle{Standard}
{\centering\selectlanguage{english}\bfseries
Hands-on project : Experiment on GOLEM
\par}
{\centering\selectlanguage{english}
R. Guirlet on behalf of the French lecturers
\par}
\bigskip
{\selectlanguage{english}\bfseries
Preamble}
{\selectlanguage{french}
\foreignlanguage{english}{This document was initially composed for students having some experience in basic tokamak
physics and experiments. It aimed at helping them to relate their academic lectures to a real tokamak environment and
particularly the GOLEM environment. It has been adapted to suit better the participants of the ASPNF School in Bangkok.
}}
{\selectlanguage{english}
A short description of GOLEM is given in part 1 and a list of questions is given in part 2. You will see that answers
are given directly below the questions. Nevertheless, you will learn much by thinking of the question before looking at
the answer! For those who did not have a specific course on tokamak physics, the vocabulary and the ideas may be new.
We will have some time during the School for group discussions on this document and on GOLEM experiments.}
{\selectlanguage{english}
Although we took some care preparing this document, you may find mistakes or typos. You are welcome to indicate them to
the coordinators of your discussion group.}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{1. What is GOLEM?}}}
{\selectlanguage{french}
\foreignlanguage{english}{GOLEM (}\url{http://golem.fjfi.cvut.cz/}\foreignlanguage{english}{) is one of the very first
tokamaks and the oldest tokamak in operation in the world. It started its carreer as TM1 at the Kurchatov Institute in
Moscow in the early 60's. It was moved to the Prague Institute of Plasma Physics in 1977, where it was operated under
the name of CASTOR until 2006. It was then moved again to the Czech Technical University (CTU) in Prague where it was
renamed GOLEM, a parabolic reference to a legend about a powerful creature made by a rabbine in Prague in order to
serve and protect the Jewish community (}\url{https://en.wikipedia.org/wiki/Golem}\foreignlanguage{english}{).}}
{\selectlanguage{english}
GOLEM was installed, commissioned and is continuously upgraded by Vojtech Svoboda with the aim of training students and
young physicists interested in thermonuclear fusion research. This is done both by allowing CTU students to develop new
systems or diagnostics for GOLEM and by organising remote experiments with groups in various places around the world
[V. Svoboda et al., Fusion Eng. Design 86 (2011) 1310-1314].}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{1.1. Characteristics of the tokamak}}}
{\selectlanguage{french}
The main parameters of GOLEM are listed in the table below:}
\begin{center}
\tablefirsthead{}
\tablehead{}
\tabletail{}
\tablelasttail{}
\begin{supertabular}{|m{1.6913599in}|m{0.53235984in}|}
\hline
{\selectlanguage{english} Plasma major radius} &
{\selectlanguage{english} 40 cm}\\\hline
{\selectlanguage{english} Plasma minor radius} &
{\selectlanguage{english} 8.5 cm}\\\hline
{\selectlanguage{french} \foreignlanguage{english}{Max. toroidal field}} &
{\selectlanguage{english} 0.8 T}\\\hline
{\selectlanguage{english} Max. plasma current} &
{\selectlanguage{english} 10 kA}\\\hline
{\selectlanguage{english} Typical plasma duration} &
{\selectlanguage{english} 15 ms}\\\hline
{\selectlanguage{english} Working gas} &
{\selectlanguage{french} \foreignlanguage{english}{H}\foreignlanguage{english}{\textsubscript{2}}}\\\hline
\end{supertabular}
\end{center}
\bigskip
{\selectlanguage{english}
The plasma cross section is circular.}
{\selectlanguage{english}
The vacuum vessel is made of stainless steel. It is usually baked with a series of cycles at 200°C before an experiment
and is operated at room temperature. }
{\selectlanguage{english}
As an example of the recent developments, the machine has been equipped with a high temperature superconducting poloidal
coil, which is still in test.}
{\selectlanguage{english}
With the provision that a responsible officer (namely Dr. Vojtech Svoboda) be in the tokamak surroundings for
reliability and safety reasons, operation of the tokamak can be performed entirely remotely. This can be done either
via a web interface or by secured access to the local linux server which controls the machine. The high repetition rate
allows to perform a discharge every 2-3 mn.}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{1.2. Adjustable parameters}}}
{\selectlanguage{english}
The hydrogen pressure in the vessel is monitored with the help of a pressure gauge.}
{\selectlanguage{english}
The other parameters which can be adjusted are:}
{\selectlanguage{french}
\foreignlanguage{english}{{}- the toroidal field on the axis
}\foreignlanguage{english}{\textit{B}}\foreignlanguage{english}{\textit{\textsubscript{Tor}}}\foreignlanguage{english}{;
}}
{\selectlanguage{french}
\foreignlanguage{english}{{}- the electric field at the breakdown
}\foreignlanguage{english}{\textit{E}}\foreignlanguage{english}{\textit{\textsubscript{BD}}}\foreignlanguage{english}{;
}}
{\selectlanguage{french}
\foreignlanguage{english}{{}- the electric field during the discharge
}\foreignlanguage{english}{\textit{E}}\foreignlanguage{english}{\textit{\textsubscript{CD}}}\foreignlanguage{english}{;
}}
{\selectlanguage{french}
\foreignlanguage{english}{{}- and (sometimes) the vertical magnetic field
}\foreignlanguage{english}{\textit{B}}\foreignlanguage{english}{\textit{\textsubscript{ST}}}\foreignlanguage{english}{
which allows horizontal stabilisation of the plasma. }}
{\selectlanguage{french}
\foreignlanguage{english}{Each of these quantities is controlled through a capacitor bank supplied with an adjustable
voltage (denoted
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{Tor}}}\foreignlanguage{english}{,
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{BD}}}\foreignlanguage{english}{,
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{CD}}}\foreignlanguage{english}{
and
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{ST}}}\foreignlanguage{english}{
resp.).}}
{\selectlanguage{french}
\foreignlanguage{english}{In addition to these physical quantities, it is possible to set a delay between
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{BD}}}\foreignlanguage{english}{,
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{CD}}}\foreignlanguage{english}{,
and
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{ST}}}\foreignlanguage{english}{
(i.e.
}\foreignlanguage{english}{\textit{E}}\foreignlanguage{english}{\textit{\textsubscript{BD}}}\foreignlanguage{english}{,
}\foreignlanguage{english}{\textit{E}}\foreignlanguage{english}{\textit{\textsubscript{CD}}}\foreignlanguage{english}{,
and
}\foreignlanguage{english}{\textit{B}}\foreignlanguage{english}{\textit{\textsubscript{ST}}}\foreignlanguage{english}{)
and
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{Tor}}}\foreignlanguage{english}{
(i.e. the toroidal magnetic field) onset.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{1.3. Diagnostics and measurements}}}
{\selectlanguage{english}
{\textbullet} GOLEM is equipped with a set of coils for magnetic measurements:}
{\selectlanguage{french}
\foreignlanguage{english}{{}- a coil around the transformer core for the loop voltage
(}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{loop}}}\foreignlanguage{english}{)
measurement;}}
{\selectlanguage{french}
\foreignlanguage{english}{{}- a Rogowski coil around the vessel for the total current measurement
}\foreignlanguage{english}{\textit{I}}\foreignlanguage{english}{\textit{\textsubscript{tot}}}\foreignlanguage{english}{\textit{
= I}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{\textit{ +
I}}\foreignlanguage{english}{\textit{\textsubscript{chamber}}}\foreignlanguage{english}{;}}
{\selectlanguage{english}
{}- a flux loop around the vessel in a poloidal section for the toroidal field measurement;}
{\selectlanguage{french}
\foreignlanguage{english}{{}- 4 Mirnov coils in a poloidal section inside the vessel for local magnetic field
measurements.}}
{\selectlanguage{english}
{\textbullet} A photodiode viewing a poloidal slice of the plasma through a midplane port window measures (in relative
units) the visible radiation intensity.}
{\selectlanguage{french}
\foreignlanguage{english}{{\textbullet} A fast camera can be mounted behind a window for imaging of a poloidal slice of
the plasma.}}
{\selectlanguage{english}
{\textbullet} A set of 20 aligned AXUV detectors (bolometers) for measurements of the radiated power profile.}
{\selectlanguage{english}
{\textbullet} Note that the diagnostics are maintained by students who work on them only pat time. As a consequence, the
only measurements which are always available are those of the magnetics. }
\bigskip
{\selectlanguage{english}
The measurements are stored in a database. A pulse summary with the main plasma parameters is displayed on the
experiment webpage. The data can also be retrieved as files for further analysis.}
\bigskip
\bigskip
{\selectlanguage{english}\bfseries
2. How to determine the main plasma physical quantities from the measurements?}
\bigskip
{\selectlanguage{english}\bfseries
2.1. Total current}
{\selectlanguage{french}
\foreignlanguage{english}{It can be deduced from the Rogowski coil measurement. The coil measures a permanent (i.e.
independent of time) voltage } $U_{\normalsubformula{\text{offset}}}$\foreignlanguage{english}{, which is called an
offset. It corresponds to the coil bias and has no relation with the discharge. During the discharge, there is an
additional voltage } $U_{\normalsubformula{\text{VP}}}(t)$\foreignlanguage{english}{ related to the current flowing in
the vessel and the plasma:}}
\begin{equation*}
U_i^R(t)=U_{\normalsubformula{\text{offset}}}+U_{\normalsubformula{\text{VP}}}(t)
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{where } $U_i^R(t)$ \foreignlanguage{english}{is the total voltage measured by the Rogowski
coil. Thus the meaningful voltage (i.e. the part related to the current in the vessel and the plasma) is obtained by
subtracting the offset from the measurement obtained during the discharge:}}
\begin{equation*}
U_{\normalsubformula{\text{VP}}}(t)=U_i^R(t)-U_{\normalsubformula{\text{offset}}}
\end{equation*}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{Then, as seen in the lecture on diagnostics, the quantity }
$U_{\normalsubformula{\text{VP}}}(t)$\foreignlanguage{english}{ is proportional to the time derivative of the current
flowing across the coil:}}
\begin{equation*}
U_{\normalsubformula{\text{VP}}}(t)\propto
\frac{\normalsubformula{\text{dI}}_{\normalsubformula{\text{total}}}(t)}{\normalsubformula{\text{dt}}}
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{The linearity coefficient is called a calibration factor. We will write it }
$C_I$\foreignlanguage{english}{:}}
\begin{equation*}
U_{\normalsubformula{\text{VP}}}(t)=C_I\frac{\normalsubformula{\text{dI}}_{\normalsubformula{\text{total}}}(t)}{\normalsubformula{\text{dt}}}
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{\ For the GOLEM Rogowski coil, the calibration factor is known:} $C_I=2\times
10^{-4}$\foreignlanguage{english}{.}}
{\selectlanguage{french}
\foreignlanguage{english}{To determine the total current }
$I_{\normalsubformula{\text{total}}}(t)$\foreignlanguage{english}{, we just have to perform an integration over time:
}}
\begin{equation*}
I_{\normalsubformula{\text{total}}}(t)=\frac 1{C_I}\overset t{\underset 0{\int
}}U_{\normalsubformula{\text{VP}}}(t')\normalsubformula{\text{dt}}'=\frac 1{C_I}\overset t{\underset 0{\int
}}\left(U_i^R(t)-U_{\normalsubformula{\text{offset}}}\right)\normalsubformula{\text{dt}}'
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{This is a theoretical formula: in reality, the measurement is not continuous. The system
performs a series of measurements separated by a small time interval } $\mathit{\Delta t}$\foreignlanguage{english}{.
}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ How do we have to adapt the theoretical formula to obtain the current?}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{In practice, to determine the total current, we replace the integral by a sum,
assuming that the current does not change during narrow time intervals:}}}
\begin{equation*}
\begin{matrix}I_{\normalsubformula{\text{total}}}(t)&=&\frac 1{C_I}\overset t{\underset 0{\int
}}\left(U_i^R(t')-U_{\normalsubformula{\text{offset}}}\right)\normalsubformula{\text{dt}}'\\&\approx &\frac
1{C_I}\overset{t/\mathit{\Delta t}}{\underset{j=0}{\sum
}}\left(U_i^R(t_j)-U_{\normalsubformula{\text{offset}}}\right)\mathit{\Delta t}\\&\approx &\frac
1{C_I}\left(\overset{t/\mathit{\Delta t}}{\underset{j=0}{\sum }}U_i^R(t_j).\mathit{\Delta
t}\right)-U_{\normalsubformula{\text{offset}}}t\end{matrix}
\end{equation*}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ How to determine }
$U_{\normalsubformula{\text{offset}}}$\foreignlanguage{english}{?}}
\bigskip
{\selectlanguage{english}\itshape
The offset is the bias of the coil before the plasma starts, which corresponds approximately to the first 450
measurement points:}
\bigskip
\begin{equation*}
U_{\normalsubformula{\text{offset}}}=\frac{\overset{450}{\underset{i=1}{\sum }}U_i^R}{450}
\end{equation*}
\bigskip
{\selectlanguage{english}\bfseries
2.2. Plasma current}
{\selectlanguage{english}
Due to the fact that the vessel is metallic, the current induced in the tranformer after the breakdown flows both
through the plasma and through the vessel. The system can be seen as the following electrical circuit:}
[Warning: Draw object ignored][Warning: Draw object ignored][Warning: Draw object ignored][Warning: Draw object
ignored][Warning: Draw object ignored][Warning: Draw object ignored][Warning: Draw object ignored][Warning: Draw object
ignored][Warning: Draw object ignored][Warning: Draw object ignored]
\begin{figure}
\centering
\begin{minipage}{0.3602in}
{\centering\selectlanguage{french}
\textit{L}\textit{\textsubscript{V}}
\par}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\begin{minipage}{0.5236in}
{\centering\selectlanguage{french}
\textit{L}\textit{\textsubscript{P}}\textit{(t)}
\par}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\begin{minipage}{0.552in}
{\selectlanguage{french}
\textit{U}\textit{\textsubscript{loop}}}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\begin{minipage}{0.7228in}
{\centering\selectlanguage{french}
\textit{R}\textit{\textsubscript{P}}\textit{(t)}
\par}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\begin{minipage}{0.7228in}
{\centering\selectlanguage{french}
\textit{R}\textit{\textsubscript{V}}
\par}
\end{minipage}
\end{figure}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{where the plasma and the vessel are in parallel. On this sketch, the upper branch represents
the plasma and the lower branch represents the vessel. } $R_P(t)$\foreignlanguage{english}{ and }
$R_V$\foreignlanguage{english}{ are the plasma and the vessel resistivities and } $L_P(t)$\foreignlanguage{english}{
and } $L_V$\foreignlanguage{english}{ are the plasma and the vessel inductances respectively. }
$R_V$\foreignlanguage{english}{ and } $L_V$\foreignlanguage{english}{ are characteristic of the vessel and thus
independent of time. Note that the loop voltage } $U_{\normalsubformula{\text{loop}}}(t)$\foreignlanguage{english}{ is
measured directly.}}
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Write down the circuit equations i.e. the expressions of }
$U_{\normalsubformula{\text{loop}}}(t)$ \foreignlanguage{english}{and }
$I_{\normalsubformula{\text{total}}}(t)$\foreignlanguage{english}{.}}
\bigskip
{\selectlanguage{english}\itshape
The circuit equations are:}
\bigskip
{\centering\selectlanguage{french}
$\begin{matrix}U_{\normalsubformula{\text{loop}}}(t)=R_VI_V(t)+L_V\frac{\normalsubformula{\text{dI}}_V}{\normalsubformula{\text{dt}}}=R_P(t)I_P(t)+L_P(t)\frac{\normalsubformula{\text{dI}}_P}{\normalsubformula{\text{dt}}}(t)\hfill\null
\\I_{\normalsubformula{\text{tot}}}(t)=I_V(t)+I_P(t)\hfill\null \end{matrix}\hfill
$\foreignlanguage{english}{\textit{.}}
\par}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ How can the plasma current be deduced from the total current and the other
available quantities?}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{It is possible to perform discharges without plasma (e.g. with no gas injected into
the vessel). In such discharges,
I}}\foreignlanguage{english}{\textit{\textsubscript{tot}}}\foreignlanguage{english}{\textit{ =
I}}\foreignlanguage{english}{\textit{\textsubscript{V}}}\foreignlanguage{english}{\textit{. Therefore,
I}}\foreignlanguage{english}{\textit{\textsubscript{tot}}}\foreignlanguage{english}{\textit{ and
U}}\foreignlanguage{english}{\textit{\textsubscript{loop}}}\foreignlanguage{english}{\textit{ being known,
R}}\foreignlanguage{english}{\textit{\textsubscript{V}}}\foreignlanguage{english}{\textit{ and
L}}\foreignlanguage{english}{\textit{\textsubscript{V}}}\foreignlanguage{english}{\textit{ can be determined.}}}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{In subsequent discharges with plasma,
I}}\foreignlanguage{english}{\textit{\textsubscript{V}}}\foreignlanguage{english}{\textit{ can then be deduced from
U}}\foreignlanguage{english}{\textit{\textsubscript{loop}}}\foreignlanguage{english}{\textit{ and subtracted from
I}}\foreignlanguage{english}{\textit{\textsubscript{tot}}}\foreignlanguage{english}{\textit{ to obtain
I}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{\textit{.}}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Using Ohm's law
(}\url{https://en.wikipedia.org/wiki/Ohm%27s_law}\foreignlanguage{english}{) applied only to the plasma branch, you will also determine the plasma resistivity.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{Once
I}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{\textit{ is known, the plasma
resistivity
R}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{\textit{${\times}$2${\pi}$R}}\foreignlanguage{english}{\textit{\textsubscript{0}}}\foreignlanguage{english}{\textit{
(R}}\foreignlanguage{english}{\textit{\textsubscript{0}}}\foreignlanguage{english}{\textit{ being the major radius) can
be obtained using the relation:}}}
\begin{equation*}
R_P(t)=\frac{U_{\normalsubformula{\text{loop}}}(t)}{I_P(t)}
\end{equation*}
\bigskip
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.3. Toroidal magnetic field}}}
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Using the same principle as for the total current and replacing } $U_i^R$
\foreignlanguage{english}{with the appropriate voltage } $U_B$ \foreignlanguage{english}{and
C}\foreignlanguage{english}{\textsubscript{I}}\foreignlanguage{english}{ with
C}\foreignlanguage{english}{\textsubscript{B}}\foreignlanguage{english}{ = 1/170, you will determine the toroidal
magnetic field.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{As the Rogowski coil, the flux coil measures a permanent (i.e. independent of time)
offset voltage }} $U_{B,\normalsubformula{\text{offset}}}$\foreignlanguage{english}{\textit{. It corresponds to the
coil bias and has no relation with the discharge. During the discharge, there is an additional voltage }}
$U_B(t)$\foreignlanguage{english}{\textit{ related to the magnetic field in the vessel and the plasma:}}}
\begin{equation*}
U_B(t)=U_{B,\normalsubformula{\text{offset}}}+U_{B,\normalsubformula{\text{VP}}}(t)
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{where }} $U_B(t)$ \foreignlanguage{english}{\textit{is the total voltage measured by
the flux coil. Thus the meaningful voltage (i.e. the part related to the magnetic field in the vessel and the plasma)
is obtained by subtracting the offset from the measurement obtained during the discharge:}}}
\begin{equation*}
U_{B,\normalsubformula{\text{VP}}}(t)=U_B(t)-U_{B,\normalsubformula{\text{offset}}}
\end{equation*}
\bigskip
{\selectlanguage{english}\itshape
The flux loop around the vessel gives a measurement of the time derivative of the magnetic field:}
\begin{equation*}
U_{B,\normalsubformula{\text{VP}}}=C_B\frac{\normalsubformula{\text{dB}}(t)}{\normalsubformula{\text{dt}}}
\end{equation*}
\bigskip
{\selectlanguage{english}\itshape
The magnetic field can be obtained by integrating this expression over time:}
\begin{equation*}
B(t)=\frac 1{C_B}\overset t{\underset 0{\int }}U_{B,\normalsubformula{\text{VP}}}(t')\normalsubformula{\text{dt}}'
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{In practice, as the Rogowski coil, the system performs a series of measurements
separated by a small time interval }} $\mathit{\Delta t}$\foreignlanguage{english}{\textit{. The practical expression
is thus:}}}
\begin{equation*}
\begin{matrix}B(t)&=&\frac 1{C_B}\overset t{\underset 0{\int
}}\left(U_B(t')-U_{B,\normalsubformula{\text{offset}}}\right)\normalsubformula{\text{dt}}'\\&\approx &\frac
1{C_B}\overset{t/\mathit{\Delta t}}{\underset{j=0}{\sum
}}\left(U_B(t_j)-U_{B,\normalsubformula{\text{offset}}}\right)\mathit{\Delta t}\\&\approx &\frac
1{C_B}\left(\overset{t/\mathit{\Delta t}}{\underset{j=0}{\sum }}U_B(t_j).\mathit{\Delta
t}\right)-U_{B,\normalsubformula{\text{offset}}}t\end{matrix}
\end{equation*}
\bigskip
\bigskip
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.4. Injected power}}}
{\selectlanguage{french}
\foreignlanguage{english}{In GOLEM, the only power injected to the plasma is by Joule effect
(}\url{https://en.wikipedia.org/wiki/Joule_heating}\foreignlanguage{english}{): as in every electrical conductor, the
plasma current and loop voltage are partly converted into heat. Do you remember P =
RI}\foreignlanguage{english}{\textsuperscript{2}}\foreignlanguage{english}{ = UI? }}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Using this formula and adapting it to the present situation, deduce the
injected power from the physical quantities determined above.}}
\bigskip
\begin{equation*}
P_{\normalsubformula{\text{inj}}}(t)=P_{\Omega }(t)=U_{\normalsubformula{\text{loop}}}(t)I_P(t)=R_P(t)I_P(t)^2
\end{equation*}
\bigskip
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.5. Electron temperature}}}
{\selectlanguage{french}
\foreignlanguage{english}{The total plasma current is the integral of the plasma current density
}\foreignlanguage{english}{\textit{j}}\foreignlanguage{english}{ over a poloidal section of the plasma:}}
\begin{equation*}
I_P=\int _{\normalsubformula{\text{Section}}}\vec j.d\vec S
\end{equation*}
\begin{equation*}
j_{\text{//}}=\sigma _{\text{//}}E_{\normalsubformula{\text{ind}}}
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{where
}\foreignlanguage{english}{\textit{${\sigma}$}}\foreignlanguage{english}{\textsubscript{//}}\foreignlanguage{english}{
is the plasma parallel \ conductivity (meaning the component parallel to the magnetic field) and
}\foreignlanguage{english}{\textit{E}}\foreignlanguage{english}{\textit{\textsubscript{ind}}}\foreignlanguage{english}{
the electric field component in the plasma current direction.}}
{\selectlanguage{french}
\foreignlanguage{english}{By replacing this expression of } $j_{\text{//}}$\foreignlanguage{english}{ in the expression
of } $I_P$\foreignlanguage{english}{, we find that the plasma current can also be written in the following way:}}
\bigskip
\begin{equation*}
I_P=\int _0^a\sigma _{\text{//}}E_{\normalsubformula{\text{ind}}}.2\mathit{\pi r}.\normalsubformula{\text{dr}}
\end{equation*}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{An expression of the parallel conductivity can be found in [J. Wesson, Tokamaks, Oxford
Science Publications, 3}\foreignlanguage{english}{\textsuperscript{rd}}\foreignlanguage{english}{ edition (2004),
Section 2.16], from which we deduce:}}
\bigskip
\begin{equation*}
I_P=1.13\times 10^3\times \frac{U_{\normalsubformula{\text{loop}}}}{2\mathit{\pi R}_0}\frac
1{Z_{\normalsubformula{\text{eff}}}}\int _0^aT_e(r)^{3/2}2\mathit{\pi r}.\normalsubformula{\text{dr}}
\end{equation*}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{where
}\foreignlanguage{english}{\textit{I}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{
is in A,
}\foreignlanguage{english}{\textit{U}}\foreignlanguage{english}{\textit{\textsubscript{loop}}}\foreignlanguage{english}{
in V,
}\foreignlanguage{english}{\textit{T}}\foreignlanguage{english}{\textit{\textsubscript{e}}}\foreignlanguage{english}{
in eV and the induced electric field has been expressed as a function of the loop voltage (note that, due to the lack
of information about the local electric field, we assume here a uniform electric field).}}
{\selectlanguage{french}
\foreignlanguage{english}{\ \ The temperature profile
T}\foreignlanguage{english}{\textsubscript{e}}\foreignlanguage{english}{(r) is not measured in GOLEM. We will assume a
polynomial form:}}
{\centering\selectlanguage{english}
$T_e(r)=T_{e,0}\left(1-\frac{r^2}{a^2}\right)^2$.
\par}
{\selectlanguage{french}
\foreignlanguage{english}{The only quantity which is not measured is the central temperature
}\foreignlanguage{english}{\textit{T}}\foreignlanguage{english}{\textit{\textsubscript{e,0}}}\foreignlanguage{english}{.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Using the expressions of
}\foreignlanguage{english}{\textit{I}}\foreignlanguage{english}{\textit{\textsubscript{P}}}\foreignlanguage{english}{
and
}\foreignlanguage{english}{\textit{T}}\foreignlanguage{english}{\textit{\textsubscript{e}}}\foreignlanguage{english}{\textit{(r)}}\foreignlanguage{english}{,
you will determine the central temperature (in eV) as a function of the measured quantities (in SI units).}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{The integral can be calculated analytically and we obtain the central temperature (in
eV) as a function of the measured quantities (in SI units):}}}
\bigskip
{\centering\selectlanguage{french}
$T_{e,0}=\left(\frac 8{1.13\times
10^3}\frac{R_0}{a^2}Z_{\normalsubformula{\text{eff}}}\frac{I_P}{U_{\normalsubformula{\text{loop}}}}\right)^{2/3}$\foreignlanguage{english}{\textit{.}}
\par}
\bigskip
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.6. Electron density}}}
{\selectlanguage{english}
As the diagnostic for density measurements (interferometer) is not always available and reliable, it can be useful to
have another method to determine the plasma density. }
{\selectlanguage{french}
\foreignlanguage{english}{We will assume that the gas injected in the vessel prior to the discharge is not adsorbed in
the vessel wall, and that it is completely ionised (this can be justified a posteriori by the high value of the central
temperature compared with the H ionisation potential). In addition, we assume that the plasma is an ideal gas, so that
the plasma ion (or electron) density is the same as the injected gas density. }}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Write the ideal gas law
(}\url{https://en.wikipedia.org/wiki/Ideal_gas_law}\foreignlanguage{english}{) and explain the physical quantities:}}
\begin{equation*}
\normalsubformula{\text{PV}}=\normalsubformula{\text{Nk}}_BT
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{where P is the pressure, V is the volume of gas, N the corresponding number of
molecules, k}}\foreignlanguage{english}{\textit{\textsubscript{B}}}\foreignlanguage{english}{\textit{ is the Boltzmann
constant and T is the gas temperature. }}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ Determine the electron density using the appropriate assumptions.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{Notice that N/V is the average molecule density. In GOLEM we work with hydrogen
(H}}\foreignlanguage{english}{\textit{\textsubscript{2}}}\foreignlanguage{english}{\textit{). If we assume that all
molecules are dissociated and ionised, each molecule will contribute 2 electrons to the total density. The average
electron density is thus obtained from the ideal gas law:}}}
\bigskip
\begin{equation*}
n_{e,\normalsubformula{\text{av}}}=2\frac{p_{\normalsubformula{\text{vessel}}}}{k_BT_{\normalsubformula{\text{vessel}}}}
\end{equation*}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{where
p}}\foreignlanguage{english}{\textit{\textsubscript{vessel}}}\foreignlanguage{english}{\textit{ and
T}}\foreignlanguage{english}{\textit{\textsubscript{vessel}}}\foreignlanguage{english}{\textit{ are the pressure and
the temperature in the vacuum vessel before the discharge.}}}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{Note that this does not take into account the impurity contribution. For a given gas
pressure, a mixture of hydrogen with other gasses will likely produce more electrons than pure hydrogen, since an
impurity atom will provide at least (and in general more than) one electron. The plasma electron density calculated as
above is thus underestimated. }}}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{NB: Assuming a parabolic density profile of the form }}
$n_e(r)=n_{e,0}\left(1-\frac{r^2}{a^2}\right)$\foreignlanguage{english}{\textit{, it is easy to calculate the relation
between n}}\foreignlanguage{english}{\textit{\textsubscript{e,av}}}\foreignlanguage{english}{\textit{ and
n}}\foreignlanguage{english}{\textit{\textsubscript{e,0}}}\foreignlanguage{english}{\textit{: }}
$n_{e,\normalsubformula{\text{av}}}=\frac{n_{e,0}} 4$\foreignlanguage{english}{\textit{.}}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.7. Safety factor}}}
{\selectlanguage{french}
\foreignlanguage{english}{The safety factor (in general denoted
}\foreignlanguage{english}{\textit{q}}\foreignlanguage{english}{) is the number of toroidal turns of a field line
necessary to complete one poloidal turn. Any two field lines on the same magnetic surface have the same safety factor,
so }\foreignlanguage{english}{\textit{q}}\foreignlanguage{english}{ is defined for each magnetic surface. In the large
aspect ratio approximation, the safety factor can be expressed as:}}
\begin{equation*}
q(r)=\frac{\normalsubformula{\text{rB}}_{\normalsubformula{\text{Tor}}}}{\normalsubformula{\text{RB}}_{\normalsubformula{\text{pol}}}}
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{In this expression, }\foreignlanguage{english}{\textit{r}}\foreignlanguage{english}{ is the
minor radius of the magnetic surface, } $B_{\normalsubformula{\text{Tor}}}(r)$ \foreignlanguage{english}{and }
$B_{\normalsubformula{\text{Pol}}}(r)$\foreignlanguage{english}{ are the toroidal and poloidal magnetic field
components averaged over the magnetic surface, and }\foreignlanguage{english}{\textit{R}}\foreignlanguage{english}{ is
the major radius of the considered surface. The only unknown in this expression is }
$B_{\normalsubformula{\text{Pol}}}(r)$\foreignlanguage{english}{. All the other quantities are measured. }}
\bigskip
{\selectlanguage{english}
${\rightarrow}$ Reminder on Ampère's law}
{\selectlanguage{french}
\foreignlanguage{english}{\ (}\href{https://en.wikipedia.org/wiki/Ampère%27s_circuital_law}{\textstyleInternetlink{\foreignlanguage{english}{https://en.wikipedia.org/wiki/Amp\%C3\%A8re\%27s\_circuital\_law}}}\foreignlanguage{english}{) : denoting }\foreignlanguage{english}{\textit{I(r)}}\foreignlanguage{english}{ the current flowing through an electrical conductor of radius r and } $\vec B$\foreignlanguage{english}{ the magnetic field, recall the expression of Ampère's law. In this expression, an integral appears over a closed path. Explain the shape of this loop.}}
\bigskip
{\selectlanguage{english}\itshape
The general expression of Ampère's law is:}
\begin{equation*}
\mu _0I(r)=\oint \vec B.d\vec l
\end{equation*}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{The integral is over a loop enclosing the electrical conductor. The scalar product
means that only the component of }} $\vec B$\foreignlanguage{english}{\textit{ along the loop will play a role.}}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ In order to determine }
$B_{\normalsubformula{\text{Pol}}}(r)$\foreignlanguage{english}{, apply Ampère's law to the case of a tokamak plasma.
In that case, }\foreignlanguage{english}{\textit{r}}\foreignlanguage{english}{ is the minor radius of the considered
magnetic surface and the integral is over a loop of the same minor radius.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{Let us apply Ampère's law to a closed poloidal loop encircling the magnetic axis at a
distance r:}}}
{\centering\selectlanguage{french}
$\mu _0I(r)=\oint \vec B.d\vec l=\oint B_{\normalsubformula{\text{pol}}}.\normalsubformula{\text{dl}}=2\pi
\normalsubformula{\text{rB}}_{\normalsubformula{\text{pol}}}(r)$ $ $\foreignlanguage{english}{\textit{,}}
\par}
{\selectlanguage{english}\itshape
where I(r) is the plasma current enclosed by the loop. }
\bigskip
{\selectlanguage{english}
${\rightarrow}$ Using the previous results and the definition of the safety factor, calculate the safety factor at the
last closed flux surface.}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{We can now express q(r) replacing
B}}\foreignlanguage{english}{\textit{\textsubscript{pol}}}\foreignlanguage{english}{\textit{ with its expression as a
function of I(r):}}}
{\centering\selectlanguage{french}
$q(r)=\frac{2\mathit{\pi r}^2B_{\normalsubformula{\text{Tor}}}}{\mu _0I(r)R}$\foreignlanguage{english}{\textit{.}}
\par}
{\selectlanguage{french}
\foreignlanguage{english}{\textit{The edge safety factor is thus: }} $q_a=\frac{2\mathit{\pi
a}^2B_{\normalsubformula{\text{Tor}}}}{\mu _0I_PR}$\foreignlanguage{english}{\textit{. It is of particular importance
in tokamak experiments since it plays an important role in the MHD stability. If the edge safety factor is close to 2,
it is almost impossible to have a stable plasma. In most tokamaks,
q}}\foreignlanguage{english}{\textit{\textsubscript{edge}}}\foreignlanguage{english}{\textit{ \~{} 3 is favourable to
plasma stability, hence the name of 'safety factor'.}}}
\bigskip
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textbf{2.8. Plasma energy content}}}
{\selectlanguage{french}
\foreignlanguage{english}{The plasma energy content can be determined using the temperature and density estimated above.
You probably remember that in a gas, temperature is defined so that the average kinetic energy of a molecule is }
$\frac 3 2k_BT$\foreignlanguage{english}{( }\url{https://en.wikipedia.org/wiki/Temperature}\foreignlanguage{english}{).
}}
{\selectlanguage{french}
\foreignlanguage{english}{In a plasma this must be refined. Instead of molecules, we have ions and electrons. In general
they do not have the same temperature, so we define the electron temperature } $T_e$\foreignlanguage{english}{ and the
ion temperature } $T_i$\foreignlanguage{english}{. These temperatures are not uniform in the plasma but they are
uniform on a magnetic surface, so we must consider their radial profiles } $T_e(r)$\foreignlanguage{english}{ and }
$T_i(r)$\foreignlanguage{english}{.}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ What are the electron energy and ion densities on a magnetic surface of minor
radius }\foreignlanguage{english}{\textit{r}}\foreignlanguage{english}{?}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{\ \ On a magnetic surface of radius r, let us denote the electron density}}
$n_e(r)$\foreignlanguage{english}{\textit{ and the ion density }} $n_i(r)$\foreignlanguage{english}{\textit{. The
electron and ion energy densities are }} $n_e(r)k_BT_e(r)$\foreignlanguage{english}{\textit{ and }}
$n_i(r)k_BT_i(r)$\foreignlanguage{english}{\textit{ respectively. }}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ What is the elemental energy in a small volume
}\foreignlanguage{english}{\textit{dV}}\foreignlanguage{english}{ around a magnetic surface of minor radius
}\foreignlanguage{english}{\textit{r}}\foreignlanguage{english}{?}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{The number of electrons in a small volume dV around a magnetic surface is }}
$n_e(r)\times \normalsubformula{\text{dV}}$\foreignlanguage{english}{\textit{. In the same way, the number of ions is
}} $n_i(r)\times \normalsubformula{\text{dV}}$\foreignlanguage{english}{\textit{. Thus, the kinetic energy contained in
this volume is }}
$n_e(r)k_BT_e(r)\normalsubformula{\text{dV}}+n_i(r)k_BT_i(r)\normalsubformula{\text{dV}}=k_B\left[n_e(r)T_e(r)+n_i(r)T_i(r)\right]\normalsubformula{\text{dV}}$\foreignlanguage{english}{\textit{
. }}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{${\rightarrow}$ What is the total kinetic energy content of the plasma?}}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{The total plasma energy is thus the sum of the elemental energy over all magnetic
surfaces:}}}
\bigskip
{\centering\selectlanguage{french}
$W_k=\int
_{V_P}n\left(\normalsubformula{\text{kT}}_e+\normalsubformula{\text{kT}}_i\right)\normalsubformula{\text{dV}}\approx
2\int _{V_P}n_e\normalsubformula{\text{kT}}_e\normalsubformula{\text{dV}}$\foreignlanguage{english}{\textit{\ \ (with n
= n}}\foreignlanguage{english}{\textit{\textsubscript{e}}}\foreignlanguage{english}{\textit{ =
n}}\foreignlanguage{english}{\textit{\textsubscript{i}}}\foreignlanguage{english}{\textit{ and assuming
T}}\foreignlanguage{english}{\textit{\textsubscript{i}}}\foreignlanguage{english}{\textit{ ${\approx}$
T}}\foreignlanguage{english}{\textit{\textsubscript{e}}}\foreignlanguage{english}{\textit{)}}
\par}
\bigskip
{\selectlanguage{english}\itshape
With the same polynomial forms as above for the density and temperature profiles, the plasma energy writes:}
{\centering\selectlanguage{french}
$W_k(t)\approx \pi
^2a^2\normalsubformula{\text{Rn}}_{e,0}(t)\normalsubformula{\text{kT}}_{e,0}(t)$\foreignlanguage{english}{\textit{.}}
\par}
\bigskip
{\selectlanguage{french}
\foreignlanguage{english}{\textit{with
W}}\foreignlanguage{english}{\textit{\textsubscript{k}}}\foreignlanguage{english}{\textit{ and
kT}}\foreignlanguage{english}{\textit{\textsubscript{e,0}}}\foreignlanguage{english}{\textit{ in J and
n}}\foreignlanguage{english}{\textit{\textsubscript{e,0}}}\foreignlanguage{english}{\textit{ in
m}}\foreignlanguage{english}{\textit{\textsuperscript{{}-3}}}\foreignlanguage{english}{\textit{.}}}
\bigskip
{\selectlanguage{french}
\textbf{\textit{\textcolor{red}{\ \ }}}\foreignlanguage{english}{\textit{There is also a way to determine the plasma
energy content from the magnetic measurements using the pressure equilibrium equation and Ampère's law. This method is
used to give the value shown on the GOLEM shot results page.}}}
\bigskip
\bigskip
\bigskip
\end{document}
