\def\GWPincludegraphics#1{\includegraphics[width=0.5\textwidth]{/GW#1}}


\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/NIMROD_87009_23_Surface1.png}
\caption{3D model of m/n = 2/1 magnetic island. \\ 
Source: \url{http://www.vacet.org/gallery/images\_video/NIMROD\_87009\_23\_Surface1.png}
}
\end{figure}
\subsubsection{Motivation}
Measurement of magnetic fields is one of the most widespread means to
study behavior of magnetically confined plasma (surprisingly). Not only
magnetic diagnostics provide global plasma parameters such as plasma
current ($I_P$), poloidal and toroidal magnetic field flux ($\psi$ and $\chi$
respectively), flux surface and plasma position reconstruction
(including real-time reconstruction for feedback control), plasma
diamagnetism (representing energy deposited in plasma) etc., but also
enable us to detect local structures presented as inhomogeneities in
current distribution. The latter case is characteristic of magnetic
islands, which represent perturbation of nested $\psi$ flux surfaces. Their
growth and rotational slowing down has serious consequences for plasma
confinement (i.e.~leads to disruption). Therefore, control of low mode
number magnetic islands, especially by electron cyclotron resonant wave
heating - ECRH, is presently hot topic among fusion community. There are
two most widespread methods on how to detect and locate these structures
in tokamak plasma, so that they might be mitigated by localized heating.
One is by SXR tomography and second is by measurement of local and
global magnetic fields induced by plasma. Within scope of this task,
measurement of local and global magnetic fields will be exploited to
study and characterize magnetic islands present during tokamak GOLEM
discharges.

\subsubsection{Theoretical Introduction}

It is advised to look into ref. {[}1{]}, or literature of similar
character to get some more detailed information on following terms.
Seventh chapter (partially the sixth one as well) of publication {[}1{]}
provides both basic and in-depth insight into character of tokamak
instabilities, especially Mirnov instabilities and tearing modes are
relevant.

Magnetic islands emerge on flux surfaces, where q safety factor (inverse
rotational transform) is of low-order m/n value - most common 2/1, 3/1,
3/2, 4/1. For GOLEM plasma, q increases with radius from plasma center
r. It can be shown that this dependency may be:


$$q(r,\nu)=\frac{2\pi B_T}{R\mu_0I_p}\frac{r^2}{1-(1-\frac{r^2}{a^2})^{\nu+1}}$$

%\begin{figure}[htbp]
%\centering
%\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/eq_2.PNG}
%\caption{}
%\end{figure}

It is evident that radius of say, q = 2/1 depends on global discharge
parameters of GOLEM - $B_T$ toroidal field and $I_P$ plasma current. Both
of these change during the experiment and thus are necessary to be
measured. See below.

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/basic_10573.png}
\caption{}
\end{figure}

This is done by magnetic diagnostics as well and will be thus in the
scope of this task. Using previous relation will enable to specify
radius of island occurence - upon specification of m/n, using q(r) = m/n
condition will yield r.

m of the island (it is safe to assume that on GOLEM only n = 1 islands
emerge), can be specified from its structure i.e.~how it behaves on
magnetics signal - most typical is as 1-20 kHz oscillation of poloidal
magnetic field, see figure below.

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/plot_10573_coil_16.png}
\caption{}
\end{figure}

When such oscillation is observed, it is safe to assume that it is due to a
magnetic island. Such oscillations are best visible when island is on
radius close to plasma edge. Explanations above imply that this is for
specific values of $B_T$ and $I_P$, ergo for specific time of discharge.
Spectrogram below shows that island is present between 16th and 19th ms.

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/plot_10573_fft_06.png}
\caption{}
\end{figure}

Doing fast-fourier transform (FFT) over this time (see below) will yield
that frequency of this oscillation is around 3 kHz.

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/plot_10573_MHD_01.png}
\caption{}
\end{figure}

If signal is measured on distribution of local magnetic field sensors as
shown below:

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/ring_coils.png}
\caption{}
\end{figure}

then the phase of oscillations will be shifted between the different
sensors, due to structure of island shown in figure on the top of this
page. Number of oscillation maxima for given time moment is equal to m
mode number that is sought. This can be specified by taking window of
signal from all the sensors simultaneously and make cross-correlation
analysis. How this is done will be covered in course. The result for
this specific island is shown below:

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/plot_10573_corr.png}
\caption{}
\end{figure}

This yields m = 3 and thus island rotates with frequency f = 3/m = 1 kHz
(taking FFT analysis from above into account). Using the equation for
q(r) would give that island is almost at the edge of plasma, hence the
nice signal.

\subsubsection{Equipment}

Poloidal flux loop, Rogowski coils, 4 Mirnov coils for local B\_theta
measurement and set of 16 Mirnov coils for the same purpose on the fig.
below.

\begin{figure}[htbp]
\centering
\GWPincludegraphics{/Education/ExperimentMenu/MHDactivity/figs/ring.png}
\caption{}
\end{figure}