Masterproef T803 : Quadratic eigenvalue problems from flowing plasma states
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Begeleiding:
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Onderzoeksgroep:
Wiskunde/Plasma Astrofysica en
Numerieke Approximatie en Lineaire Algebra Groep
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Context:
The theory to describe all linear waves and instabilities, given an exact stationary, gravitating, magnetized plasma equilibrium, in any dimensionality, has been known since 1960, and is the Frieman-Rotenberg equation. The equation governs the spatio-temporal variation of a Lagrangian fluid element displacement. When one is interested in eigenfrequencies, the temporal variation can be fixed to be exponential, and eigenfrequencies are found for all solutions obeying the Frieman Rotenberg equation and suitable boundary conditions. In the case of a static (no flow) equilibrium, the eigenvalue determination is governed by a self-adjoint operator in the Hilbert space of solutions. Due to the self-adjointness, only real eigenvalues can occur, so that stable waves or instabilities result. The full (mathematical) power of spectral theory governing physical eigenmode determination comes into play when using the Frieman-Rotenberg equation for moving plasma states. |
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Doel:
In that case, a generalized operator involving flow terms, together with a Doppler-Coriolis operator appear in the quadratic eigenvalue problem. Both operators are self-adjoint, but allow the possibility of intrinsically complex eigenvalues, with overstable modes e.g. driven by shear flow (Kelvin-Helmholtz). Unlike the case of static plasmas, where the transition to instability in the complex eigenfrequency plane can only happen through the marginal zero frequency, the eigenmodes of stationary plasmas may enter the complex eigenfrequency plane from different locations along the real oscillation frequency-axis. The master thesis research will use existing codes exploiting finite-element discretizations to numerically translate the eigenvalueeigenfunction determination, to be solved with modern numerical linear algebra techniques. The determination of eigenvalues-eigenfunctions for accretion disk configurations will then be confronted with recent theory, allowing to classify the eigenmodes in the complex eigenfrequency plane. |
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Relevante literatuur:
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Profiel:
Interest in numerical linear algebra, as well as numerical fluid dynamics. Deze masterproef is voor 1 student. |