Algebraic description of the tomography problem

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 Mistral device

General description

Let us consider a laboratory plasma device using a DC discharge to create a plasma from a hot electron beam (picture and general schema Figure 1.3), and let us call it Mistral [Matsukuma03]. Mistral has be conceived by Thiéry Pierre and Gérard Lecrert based on the Mirabelle device at the Laboratoire de Physique des Milieux Ionisés et Applications (Nancy 1982). The primary objective is to study low frequency instabilities (typical frequencies lower than the ion cyclotron frequency) and the associated transport in weakly magnetized plasmas.
Mistral consists of two main parts. The first is the source chamber con-taining the 32 tungsten filaments (the cathode) powered by 3:4 A each to emit hot electrons, or primary electron, following the Richardson law (a 1. Another energy source technology helped by plasma physics are the solar panels. They benefit from plasma surface processing mentioned earlier for more efficient produc-tion and higher energy efficiency.

MISTRAL DEVICE

Heated metal will emit a current). The anode is a honeycomb-shaped grid, placed before the wall of the source chamber. It has an alternated permanent magnets pattern creating a local magnetic configuration to prevent emitted electrons from reaching the anode before creating a plasma in the source chamber or being injected into the second section: the linear interaction chamber. The plasma is created from the interaction of primary electron with neutrals creating lower energy electrons, called secondary electrons.
A metallic diaphragm (or limiter) setting the plasma column diameter separates the two sections. Two different configurations are possible for the separation, as shown in Figure 1.5. In configuration a), the plasma of the linear column does not see the grounded limiter, but only the polarisable grid, called the separating grid. In configuration b), however, the limiter is linked to the cylinder (both have the same polarisation) and is directly facing the plasma edge. Configuration b) has been set-up to check the influence of the axial boundary conditions, and for comparisons with numerical simulations. As a result, only experiments for the single sensor tomography presented in Chapter 3 were done using configuration a), the rest used configuration b). The separating grid serves as a selective filter by putting a potential barrier for the primary electron to cross. The linear chamber is 1:2 m long and 0:4 m wide, has an axial magnetic field Bz tunable from 10 to 25 mT (= 100 to 250 G 1) and is ended by the collector, the other polarisable grid setting the axial boundary limit. Figure 1.4 summarises all power supply used to polarise different elements. A 20 cm wide cylinder is placed around the plasma to also control the radial boundary condition.
Figure 1.5 – Two configurations for the separation between the linear column (left side) and the plasma source (right side).
Mistral has a cross fields configuration (E ? B), ubiquitous in tokamaks, hall effect thrusters or ion sources. The ratio kEk=kBk (kE k 100 V=m and kBk 20 mT) is comparable to the one in tokamak edges. However, Mistral is a human scale device, a single person is needed to run it. It has also been conceived with many optical apertures and accesses for intrusive diagnostics. As will be discussed later, it also has the advantage to run in steady state for hours. Its only time constraint is the increase in temperature of the source chamber that ends up heating adjacent electronics and the whole room. The self-imposed soft limit is around two to four hours, depending on the starting ambient room temperature.
1. The magnetic field is fixed by the coils current Icoils, and in this configuration we have kBz (r = 0)k [Gauss] ’ Icoils [Ampere]. So the Gauss unit will here be preferred to its S.I. counterpart.

Mistral parameters

The two tables 1.2.1 and 1.2.2 summarise values of fixed or manually controlled experimental parameters, and the associated typical plasma para-meters on Mistral, respectively.
The Larmor radii between the ions and electrons results are very different. As a result, only the electrons are magnetised, the ions are only weakly magnetised: L;e a and L;i a.

Existing diagnostics and tools

The diagnostics available on Mistral will only be briefly presented as they are largely documented in [Jaeger10; Rebont10; Lefevre11; Escarguel12]. The description given here will mainly focus on their use in the context of this work.

Electrostatic probes

Electrostatic probes, or Langmuir probes, are one of the most common diagnostic in plasma physics. They allow to calculate the plasma potential and the electron temperature and density. The principle is to have a small piece of conducting material in the plasma and to measure the collected cur-rent for different voltages (I(V) characteristics). On Mistral, radially movable cylindrical Langmuir probe are the most used. Three probes are routinely available. they are placed so that two are in the same (r; ) plane but on opposite side, and two are aligned (same ) but at different axial positions (see figure 1.3). There are three main downsides to this diagnostic: it is in-trusive, a single probe can a priori only give information on one point of the plasma, and the plasma parameters calculated are highly dependent on the model used to infer them from the I(V) characteristics (each bearing different hypotheses). On Mistral, the axial magnetic field and the primary electrons make the plasma anisotropic and bi-Maxwellian. Therefore, the Druyvestein formula [Druyvesteyn30] is used to reconstruct the electron energy distribu-tion function f from the second derivative of the I(V ) curve (current versus polarization) as follow:
in which me and e are the electron mass and charge, and S is the probe area. Densities and temperatures then come from the first and second moments of f respectively. This method has the advantages of being independent of the probe shape (as long as it is convex) and being weakly sensitive to the details of the electron velocities distribution [Claude94]. As long as the magnetic is low enough (i.e. L;i > a), the previous theory still provides adequate results. For L;e < a, the electron saturation current will be reduced, so the I(V) characteristic can not be interpreted for high values of V any more.
Electrostatic probes can also be used at a fixed voltage. The collected current is then converted to a voltage through a resistor to monitor the plasma behaviour with an oscilloscope. Usually voltages V > Vp are chosen in order to only collect electrons. 1 This also leads to an original diagnostic described here after.

Sonification of the plasma

Monitoring the plasma behaviour with an oscilloscope requires to be able to look at its screen. However when conducting an experiment one is often required to move around the room and/or control other diagnostics, for in-stance on a computer. As will be described in section 1.3, rotating modes can be studied on Mistral. With rotation frequencies of a few kHz, they allow the aforementioned probe signal to be emitted from speakers [Escarguel12]. Our ears are very sharp to detect frequency variations. 1 The plasma can then be freely monitored from anywhere in the room, even when focused on something else. This diagnostic proved extremely useful to monitor the stationarity of the rotating modes during the measurements described in Chapter 3.

Intensified camera

A 4QuickE camera is available on Mistral. It can take a standard 25 images per seconds, but its exposure time can be down to the nanosecond. It has an intensified CCD with 736 572 pixels and a spectral range on the visible wavelengths (400 – 800 nm, and an optical zoom up to 5. Pictures from the end of the column as well as from the side aperture can be taken, either directly (as shown in Figure 4.2 later) or averaged during several syn-chronised acquisitions, as shown in [Annaratone11].
When imaging the plasma from the end of the column (to have an aver-age radial distribution of the emissivity), the sensor dynamics prevents the simultaneous imaging of the core and edge plasma. If the core plasma is measured, the edge is seen as black. Conversely the core plasma has to be masked to be able to see the edge.

Photomultiplier tubes and photodiodes

Photomultiplers tube (or just photomultipliers) and photodiodes are non-intrusive (optical measurements). Contrary to the camera they only have one ’pixel’, but their field of view can be more freely controlled, 2 and since they do not require to read many pixels, they are much faster. They are usually used with an optical fibre and a spectral filter to look at a specific wavelength of a collimated line of sight. The photomultipliers and photodiodes used for the tomography diagnostics of this thesis are further described in Chapter 3.

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Spectrometers

A spectrometer only disperse light, so they are combined with optical sensor, which are usually either a CCD (can measure a whole spectral range, but is quite slow) or a photomoltiplier (can only measure one ray, but is much faster). In the following paragraphs, the sensor will implicitly be included in the spectrometer.
Four spectrometers can be used on Mistral to resolve the wavelength spectrum of the incoming light, depending on the required spectral resolu-tion, and temporal resolution. The temporal resolution is also linked to the emissivity of the plasma and the sensitivity of the sensor. Spectrometers are non-intrusive, and many parameters can be calculated directly, such as the ion/neutral temperature (through doppler broadening), electron dens-ity (through Stark broadening), and ion/neutral density (through spectrum integration), or using more advanced models.
During this work one of the spectrometers 1 have been used for two pur-poses, thanks to its broad spectral range: 400 – 980 nm. It combines three smaller spectrometers with complementing spectral range. However, its spectral resolution is quite low ( 1 nm).
First, the plasma absolute emissivity can be estimated for different ex-perimental conditions for the choice of optical sensors, as described chapter 4.
Secondly, it is a convenient tool to check the quality of the vacuum. After an operation at atmospheric pressure on the vacuum chamber, the new va-cuum created will be polluted by air (N2 and O2), and also by water desorbed by the wall. The natural heating produced by the source chamber will then help the desorption, but the first few plasmas will always be polluted. The spectrometer is used to check when the plasma is clean enough by looking for the presence of molecular rays (mainly H2O, N2 and O2) Molecular rays are much larger than atomic rays, which makes them easily identifiable.

Laser induced fluorescence

The laser induced fluorescence is a local and non intrusive diagnostic which was installed and extensively used during the PhD of C. Rebont [Re-bont10]. The principle is to excite atoms (neutrals or ionised) with a laser of known wavelength, which fluoresce through intermediate states [Hill83]. Particles directional velocities are then calculated through the shift and broadening of the measured luminescence spectrum, and local electric field can be calculated through fluid models or conservation of the particles energy.

Calibrated black body

The black body is not a direct diagnostic, but a tool associated with them. It is an integrating sphere with fixed a one inch output hole, considered as an isotropic light source. Its emissivity is controlled by a slit placed between the lamp and the sphere. A calibrated photodiode placed inside the sphere gives the spectral emissivity in µW m 2 sr 1 nm 1: how much power of light is emitted by one square centimetre of the black body and received by a sensor intercepting one steradian of the light beam, for each nanometre of the spectrum. The black body serves as a reference to calibrate all optical measurements. Figure 1.6 shows the Jaz spectrometer spectrum before and after calibration. The three different lines on the left graph represent the three channels of the spectrometer described previously. It has been a valu-able tool for this thesis as will be described in Chapter 4.

Mistral plasma emissivity

For the experimental conditions considered in this manuscript, the plasma spectrum is largely dominated by neutral argon emission lines [Escarguel10]. If we suppose that the plasma is homogeneous along the spectroscopic line with h the energy, n the density of exited atoms, and A the Einstein coefficient corresponding to the transition of interest. In Mistral, the plasma is created by energetic primary electrons (density ne,p) in addition to the thermal electron population (density ne,th).
were n0 is the neutral density, and h viT0e,p and h viT0e,p are the temperature dependant rate coefficients for excitation from the ground state to the n state by thermal and primary electron collisions respectively.
So the measured emissivity is proportional to the primary and thermal elec-tron densities, with the cross sections as ratios (which depends on their re-spective temperatures). Their relative contribution to E is proportional to the ratio b=a. In the experimental conditions of Mistral, it is shown that b ’ 103 a [Escarguel10].

Coherent rotating modes

Coherent rotating modes have been arousing interest for half a century [Vlasov65], yet they are still not fully understood. In fact, it is a generic term regrouping different mechanisms and/or different instabilities. In this section, basic knowledge on coherent rotating modes are given, based on the different approach found in the literature, before focusing on the specific case of Mistral.
1. There is no detectable emissivity due to bremsstrahlung in Mistral because of the plasma conditions [Griem97].

General description

Two classes of coherent rotating modes are usually distinguished, depend-ing on their parallel wave vector kk = k:B=B: ‘Flute modes’ 1 for kk = 0 and ‘screw modes’ for kk > 0 (usually linked to drift waves instabilities). Both appear in plasmas with cross-field configurations (E ? B). On Mistral the axial wave number have been measured to be at least much smaller than 2 =L, with L the length of the column, hinting at flute modes. Modes with kk > 0 tend to appear at higher magnetic fields, as the ion cyclotron fre-quency affects the instability [Jaeger10].
Flute modes and screw modes are based on a perturbation of the density n and electric potential ’. The phase shift (n; ’) between the two per-turbations is critical for the growth of the instability, as illustrated in Figure 1.7. In the case of a pure drift wave, (n; ’) 0 [ ], the E B particle flux is zero and the initial perturbation does not grow. In contrast, when (n; ’) = =2, which is typical of flute modes, the E B fluxes acts to reinforce the initial perturbation and the growth is maximum.

Existing work

World

Coherent rotating structures have been frequently observed in various lin-ear devices [Fredriksen03; Manz11; Kobayashi11; Cortazar15]. These devices usually operate at plasma densities and magnetic field higher than on Mis-tral, but the experimental conditions for the observation of rotating modes and their frequencies tend to be similar. The interpretation of the coherent modes tends, however, to be quite different. Namely, some coherent low-frequency rotation modes are explained as flute modes [Brochard05] or as a generic E B drift instability [Smolyakov13], some can be intermittent [Antar07] or in steady state [Gravier04]. In some cases, the rotating plasma arm(s), expelled from the core plasma, is also associated with an ionisation front [Boeuf13], source of plasma in the edge.

Table of contents :

1 Introducing the protagonists 
1.1 Introduction to plasmas
1.1.1 Plasmas and their physical parameters
1.1.2 Natural plasmas
1.1.3 Technological applications
1.2 Mistral device
1.2.1 General description
1.2.2 Mistral parameters
1.2.3 Existing diagnostics and tools
1.2.4 Mistral plasma emissivity
1.3 Coherent rotating modes
1.3.1 General description
1.3.2 Existing work
2 Tomography 
2.1 Algebraic description of the tomography problem
2.1.1 Finite volumes scheme
2.1.2 Finite elements scheme
2.2 Tomography inversion
2.3 Numerical tomography code
2.3.1 Description
2.3.2 Benchmarking
2.4 Comments on the chords distributions
2.5 Conclusion
3 From a single sensor tomography diagnostic 
3.1 Mono-sensor tomography set-up
3.1.1 Trigger system
3.1.2 Sensors and acquisition systems
3.1.3 Optomechanical configuration
3.2 Experimental tomography acquisitions
3.3 Conclusion
4 To the 128 synchronous sensors diagnostic 
4.1 System overview
4.2 Hardware description
4.2.1 Optical sensors
4.2.2 Acquisition board
4.2.3 Optomechanical configuration
4.3 Installation
4.3.1 Collimation and alignment
4.3.2 Acquisition device check-up
4.3.3 Calibration and linearity
4.4 Experimental measurements
4.4.1 Raw signal
4.4.2 Reconstructed emissivity
4.5 Conclusion
5 Characterisation of rotating modes 
5.1 Radial and azimuthal structure of rotating modes
5.2 Parametric study of rotating modes
5.2.1 Variation of the azimuthal mode number
5.2.2 Hysteresis of the plasma
5.2.3 Radial electric field and plasma potential
5.3 Conclusion
Conclusion

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