Visualization of electron temperature and density
isosurfaces or isocurves in Tokamak plasmas
Francisco Salzedas
Dept. de Física da FEUP
Rua Dr. Roberto Frias, s/n 4200-465 Porto
fsal@fe.up.pt
and
Centro de Fusão Nuclear IST/EURATOM
Av. Rovisco Pais 1049-001 Lisboa
fsal@cfn.ist.utl.pt
ABSTRACT
In this paper are described two programs that allow the easy visualization of isosurfaces S(r,t) or
isocurves C(r,t), time and space profiles from electron temperature and density of Tokamak
plasmas, used in research of thermonuclear fusion of light nucleus.
The programs are written in IDL and they run in two European laboratories, JET (Joint European
Torus) that is the largest Tokamak in the world, based in the UK and ASDEX Upgrade a German
Tokamak, based at the IPP Max Plank Institute near Munich.
The user interface is graphical and is similar on both programs in order to facilitate its usage. A
Fourier analysis module is included to allow a quick visualization of spectrogram from channels
and compare them with isosurfaces. This analysis is very useful to detect plasma instabilities.
Any of these programs can be applied to any other kind of data with the same structure S(r,t).
INTRODUCTION
A solid that is continuously heated will, at a certain temperature, melt into liquid. If the
temperature of the liquid continues to increase the molecules that form the liquid will gain enough
energy to move away from each other and the liquid will turn into vapor. By increasing the
temperature of the gas much further, the kinetic energy of the colliding atoms of the gas will be
enough to separate electrons from their nucleus. This resulting hot "gas" of electrons moving
through positively charged nucleus is called plasma.
In plasmas made of light atoms, like Tritium, an isotope of Hydrogen, when heated up to
temperatures of a few million degrees Celsius, some nucleus will have enough kinetic energy to
overcome their mutual electric repulsion and they fuse. An enormous amount of energy is
released that is 3 orders of magnitude larger than the energy of the colliding nucleus. This is the
energy reservoir that supplies the stars through out all of the universe, like the Sun. The
controlled access to this stupendous energy reservoir is the main goal for research in Controlled
Thermonuclear Fusion.
A magnetic field of a few Tesla is used to contain such hot plasmas. The most successful
configuration nowadays is one where the magnetic field has the form of a torus. The plasma
confiner that creates such a field in a toroidal vacuum chamber is called a Tokamak (see Fig. 1).
Electron density and temperature are two of the several plasma parameters measured in
Tokamak plasmas. Both parameters mentioned in this work are measured with microwaves using
two different processes.
Measurement of the temperature is done in a passive process, by a heterodyne radiometer tuned
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transformer yoke
coils for plasma position control and
coils of primary winding transformer
B toroidal
Ip
plasma column
vacuum vessel
diagnostic port
B poloidal
poloidal coils
that create
helical field lines
B toroidal
Figure 1 – Schematic view of a Tokamak. Bpoloidal is created by the plasma current Ip.
to the range of electron cyclotron frequencies, ωc=eBφ/m emitted by electrons, with charge e and
mass m, which spiral through the strong magnetic toroidal field, Btoroidal. The spatial variation of
this field is well known and is inversely proportional to the torus major radius. This permits to
localize the radial position where a detected microwave with frequency ωc was emitted from. For
optically thick plasmas the intensity of this Electron Cyclotron Emission (IECE) can be described by
black body radiation and therefore is proportional to the electron temperature Te. The
measurement of IECE at different frequencies allows the measurement of Te at different radii. So a
heterodyne ECE radiometer (see Fig. 2a)) measures electron temperature profiles along a
horizontal chord, at different times.
Measurement of the electron density is done by a broad band reflectometer, in a active process,
by emitting microwaves into the plasma and detecting their reflection from the plasma (see Fig.
2b)). This method is far more difficult than the previous described one to measure the
temperature. In principle reflection of the microwaves should occur at a specific density layer
where the incident wave frequency matches the plasma frequency at that layer. Since the plasma
frequency is proportional to the square root of the electron density, the frequency of the reflected
wave determines the density at the reflection layer. From the time delay between the emitted and
reflected wave it is possible to determine the layer position. This procedure is rather involved and
disturbed by density fluctuations.
The two programs here described, TradView and NeRef allow to visualize and manipulate
electron and density isosurfaces, in a relatively easy way. The program TradView, runs in the two
European laboratories, where JET and ASDEX Upgrade are based. The code NeRef only runs
for ASDEX Upgrade, where a reflectometer developed by Centro de Fusão Nuclear from Instituto
Superior Técnico, operates [1].
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a)
Emission at cyclotron
frequency depends on
electron temperature
Radiometer
b)
Incident wave
reflected at critical
electron density
Reflectometer
Figure 2 – a) Schematic view of a ECE radiometer and b) of a reflectometer in a circular cross section
plasma.
ELECTRON TEMPERATURE ISOSURFACES
In the core of the program TradView are the IDL routines SHADE_SURF and CONTOUR. The
first routine creates a 3D shaded-surface representation of a regular or nearly-regular gridded
surface with hidden surfaces removed. It is possible to rotate the 3D shaded-surface about the X
- axis and about the Z - axis. The standard view in TradView has a 90º rotation about the X - axis
showing the surface as viewed from above, i.e. the observer is looking down the Z - axis. The
CONTOUR routine draws a contour plot from data stored in a rectangular array. Both line
contours and filled contour plots can be created.
The three main arguments of both SHADE_SURF and CONTOUR routines are three arrays, X, Y
and Z. X is a one dimension array (a vector) where time coordinates are stored. Y, where spatial
coordinates are stored, is a one dimension array in TradView running at ASDEX Upgrade and a
two dimension array on the other program. The reason is that at ASDEX, the position of the
measuring channels is constant with time. The Z array is a two dimensions array and stores the
electron temperature for each spatial and time coordinates. These are high-level routines with
several keywords that can be used to manipulate the surface or the contour plot. It is possible to
construct high quality plots of isosurfaces and isocurves with command-line based programs with
just a few IDL code lines [2]. However to manipulate such simple codes knowledge of IDL is
necessary.
The TradView and NeRef programs here presented are large codes with more than 2000 lines
because they use WIDGETS [3] which are simple graphical objects such as pushbuttons or
sliders that allow user interaction via a pointing device (usually a mouse) and a keyboard.
However these extra lines of code result in a friendlier user interface. IDL widgets are significantly
easier to use than other alternatives, such as writing a C language program using the native
window system directly. IDL handles much of the low-level work involved in using such toolkits.
The interpretive nature of IDL makes it easy to prototype potential user interfaces. In addition to
the user interface, the author of a program written in a traditional compiled language also must
implement any computational and graphical code required by the program. IDL widget programs
can draw on the full computational and graphical abilities of IDL to supply these components.
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Figure 3 – Screenshot of TradView showing temperature colored isosurfaces and isocurves. The cursor
coordinates (radial position, time and electron temperature in eV) are indicated. The numbered dots on the
right mark the radiometer channels position.
Figure 4 – Screenshot of TradView showing electron temperature time evolution for 3 radiometer channels.
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Figure 5 – Screenshot of TradView showing electron temperature profiles at 3 different times. The time of
the colored profile can be selected at the lower moving slide. The color scale is equal to the one used in Fig.
3. Channels position can also be over-plotted
Programs that use widgets are event driven. The program creates an interface and then waits for
messages (events) to be sent to it from the window system. Events are generated in response to
user manipulation, such as pressing a button or moving a slider. The program responds to events
by carrying out the action or computation specified by the programmer, and then waiting for the
next event.
The style of widgets IDL creates depends on the windowing system supported by the host
computer. TradView and NeRef work on Unix and Microsoft Windows systems without change.
The size of the 2D Z array containing the temperature information can easily be 50000X96 points.
With TradView the user can easily move through the data, zooming in and out. The cursor as +
form, with cross hairs extending to the edge of the plot where the cursor time, radius and
temperature are shown as it moves (see Fig. 3). The user can also remove channels that are not
good and plot only the fluctuating component of the temperature. It is possible also to visualize a
Te profile at specific time (see Fig. 4) or 1 to 3 Te superimposed profiles at different times (see
Fig. 5).
ELECTRON DENSITY ISOSURFACES
As a consequence of the measuring method, the automatically calculated electron density often
reveals degenerate profiles. Such profiles have radii coordinates with two different density values.
Such feature requires careful reprocessing. Nonetheless the global behavior of the profiles is
informative.
To overcome the obstacle of this often observed degeneracy, it is used a different IDL routine,
called PLOTS. This routine can plot a vector in three dimensions like a density profile at a certain
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Figure 6 – Screenshot of NeRef showing electron density profiles at 2 different times.
Figure 7 – Screenshot of NeRef showing electron density colored isosurfaces.
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time (see Fig. 6).
Using a keyword from the PLOTS routine called COLOR, a color profile is attributed to each point
of the density profile (vector). Normalizing the color profile to maximum and minimum values of
density measured in all the data, the value of the density is displayed in the profile both by the Z
position of the curve in the plot and by the color assigned to that Z position. Basically, to each
density value is assigned a color index.
Plotting these profiles in a reference system similar to the one used in TradView, i.e. where the
observer is looking down the Z-axis, an electron density isosurface is obtained (see Fig. 7).
The synchronized plot of electron density isosurfaces with temperature isosurfaces or other
plasma parameters shows that this average representation of the density is in agreement with
features observed in the behavior of the other parameters [4].
REFERENCES
[1] M. E. Manso 1993 Plasma Phys. Control. Fusion 35 B141
[2] F. Salzedas, S. Hokin, F. C. Schuller, et al. 2002, Phys. Plasmas, 9, 3402
[3] Research Systems Inc., 2000, IDL Reference Guides 5.4.
[4] F. Salzedas, G. Pautasso, I. Nunes, et al 2002, 29th EPS Conf. on Contr. Fusion and Plasma
Phys., Montreux, Switzerland, P1.039
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