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Concept# Kelvin–Helmholtz instability

Summary

The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. Kelvin-Helmholtz instabilities are visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres of the Sun and other stars.
Theory overview and mathematical concepts
Fluid dynamics predicts the onset of instability and transition to turbulent flow within fluids of different densities moving at different speeds. If surface tension is ignored, two fluids in parallel motion with different velocities and densities yield an interface that is unstable to short-wavelength perturbations for all speeds. However, surface tension is able to stabilize the short wavelength instability up to a threshold velocity.
If the density and velocity vary continuously in space (with the

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In the frame of this thesis, two similar high current DC arc (HCDCA) plasma sources were investigated in a low gas pressure regime (10-3-10-2 mbar). One of them was initially designed for the epitaxial growth of silicon and silicon germanium (LEP), the other for the industrial deposition of diamond (BAI). The LEP source was analysed using pure argon plasmas. Measurements of the ion saturation current were performed with a custom-built multi-Langmuir probe to analyse plasma density homogeneity. Plasma instabilities at 50 Hz were observed and studied by different means. One source of the instability is the AC current used for the filament heating, whereas lower frequency instabilities are due to the use of a ring shaped anode. A sensitive Hall sensor was used to measure the magnetic field induced by the discharge current. It was found that depending on the plasma parameters gas pressure and external magnetic field the current tends to attach at different points on the anode, leading to a complete loss of reactor symmetry. The installation of an additional cusp field around the reactor chamber lead to an increase of the overall homogeneity of the plasma density, but it could not resolve the problems of current attachment. In the following the ring anode was replaced by a point anode and the maximum external magnetic field strength was increased by a factor of ten. With a novel multi-Hall probe the current density of the now columnar shaped plasma was measured, showing a strongly peaked current density profile, when an external magnetic field is applied. Together with ion saturation current measurements made with a Langmuir probe, the electron temperature inside the plasma column was estimated to be about 4 eV. In earlier works made on the BAI reactor the high dissociation efficiency of the HCDCA plasma source has already been shown. Optical emission spectra were compared between RF plasmas, low pressure (1.5mbar) HCDCA plasmas and our very low pressure (10-3-10-2 mbar) HCDCA plasmas. The dominating species found in RF plasma spectra are molecular, the spectra of low pressure HCDCA plasmas are dominated by atomic species and the spectra of very low pressure HCDCA plasmas are dominated by ions, emphasising the high dissociation efficiency of this system. Silane and Methane was used as precursor gases in the BAI reactor for the deposition of silicon carbide films. Promising high deposition rates up to 9 nm/s were found, but FTIR spectroscopy showed high hydrogen and oxygen concentrations in the porous films, making the not optimised deposited material useless for the application of wear-resistant coatings. Microcrystalline hydrogenated silicon (µc-Si:H) is viewed as a cost- and energy-effective alternative to crystalline silicon for the production of solar cells. A detailed analysis of the deposition rate and the Raman crystallinity of µc-Si:H films deposited in the BAI reactor showed deposition rates up to 6.5 nm/s and a wide range of crystallinity from 0-80%. Film thickness inhomogeneity in silicon solar cells has to be less than 5%. To meet this standard a first substantial improvement was achieved with the installation of a linear gas injection along the plasma column. Tests on large surface glasses (47 × 37 cm2) revealed strong diffusive effects, which could be reproduced with a simple gas diffusion model. The model showed the necessity to reduce the dead volume around the plasma and to set the substrates as close as possible to the plasma column in order to minimise film thickness inhomogeneity due to diffusion. Deposition rate measurements made in these conditions assured the results of the model. The development of a method to estimate the dissociation efficiency of the plasma by simple pressure measurements showed also an important increase of the silane dissociation from 75-92% when the plasma is confined in a smaller volume. Therefore, compared to the reactor with a large dead volume, only a third of the initial silane is lost to the pumps.

It is known that the pitchfork bifurcation of Kelvin-Helmholtz instability occurring at minimum gradient Richardson number Ri(m) similar or equal to 1/4 in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, Pr. Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the 'Drazin model' base flow. Bifurcation diagrams are produced for fluids with Pr = 0.7 (typical for air), 3 and 7 (typical for water). For Pr = 3 and 7, steady billow-like solutions are found to exist for strongly stable stratification of Ri(m) beyond 1/2. Interestingly, these solutions are not a direct product of a Kelvin-Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least Pr approximate to 2.3 and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.

2021A non-field aligned coordinate system was recently implemented in GBS, a three-dimensional drift-reduced Braginskii fluid code to simulate turbulence in the plasma periphery of tokamaks. This avoids the singularity present in field-aligned coordinates at the X-point, thus allowing the simulation of any toroidally symmetric magnetic field configuration, with no separation between equilibrium and fluctuating quantities, therefore evolving self-consistently the formation of the plasma profiles. Simulations are carried out in the single-null and double-null configurations and first simulations in innovative exhaust configurations (such as the snowflake) are being performed. The talk will focus on the double-null configuration, which is of interest as a possible heat exhaust solution and for advanced heating schemes. The new insights obtained in the double-null configuration from the GBS simulations will be presented and thoroughly analysed. The different nature of the plasma dynamics in the low- and high-field sides will be pointed out, showing that turbulence is driven by a Kelvin-Helmholtz instability on the high-field side, and by an interchange instability on the low-field side. On the low-field side, a structure with two scale lengths forms. The wave-like nature of turbulence across the last closed-flux surface will be discussed and estimates of the narrow pressure scale length in this region will be given. It will be shown that blobs are generated across the last-closed flux surface, being responsible for the convective nature of the transport in the far SOL. The blob generation rate, size, and speed will be estimated, allowing the prediction of the far SOL width. Comparisons of the simulation results with previous and new analytical results will be presented, as well as with experimental results. Generalisation to other tokamak configurations will also be discussed.

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