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Minimisation problems in ideal magnetohydrodynamics
No full textThis dissertation deals with minimisation problems related to magnetohydrodynamics. The main part of this thesis is divided in 5 chapters. In the first chapter it is shown that the magnetic energy under the so called helicity constraint admits a global minimiser for each prescribed value of the helicity and that all such minimisers are Beltrami fields, i.e. eigenvector fields of the curl operator. This result is already known in the settings of compact manifolds without boundary and bounded domains with smooth boundary in . We generalise these results to the setting of abstract, compact manifolds with boundary and by this we filled a gap in the literature. In the second chapter we consider the problem of finding a domain of prescribed volume for which the minimal energy in a given helicity class becomes minimal among all other minimal energies, in the same helicity class, of domains of the same volume. This problem was considered in the literature in the setting where the ambient space is . We generalise the known results to the setting where the ambient space is any Riemannian manifold and derive a second variation inequality, which contains terms involving the geometry of the ambient space. The question of whether or not an optimal domain exists is still open and subject of current research. In the third chapter the focus lies on the field line dynamics and zero set structure of Beltrami fields, hence in particular of energy minimising vector fields. Our most important results are on the one hand the observation that the restriction of a Beltrami field to the boundary, assuming it is tangent to it, on a simply connected, compact manifold is always a gradient field, which provides us with a good understanding of the boundary field line behaviour. On the other hand we show that the Hausdorff dimension of the zero set is at most one. This upper bound is already known for the zero set in the interior of the manifold. We show that this upper bound stays valid if we include the zeros on the boundary. Such a result appears to be new in the literature. In the fourth chapter we consider again the minimisation problem from the first chapter, but add an additional symmetry constraint. Results concerning the existence of rotationally symmetric Beltrami fields on domains in are known in the literature. We generalise these results to the setting of abstract manifolds and develop new arguments to deal with general Killing vector fields. The last chapter deals with minimisation problems on . It is easy to see that the original energy functional does not admit any global minimisers if we consider it on the whole -space. That is why we consider two related energies under the helicity constraint. For the first energy it is shown, in view of Lions' "concentration compactness principle", that the only possible obstruction for the existence of global minimisers are dichotomy effects. For the second energy we derive necessary conditions for global as well as local minimisers
Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes
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Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes
No full textAsymptotic windings, surface helicity and their applications in plasma physics
Get PDFIn [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines. (ii) We prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e. at least one hole).
We then focus on toroidal surfaces which are of relevance in plasma physics and express surface helicity in terms of average poloidal and toroidal windings of the individual field lines of the underlying vector field which enables us to provide a connection between surface helicity and rotational transform. Further, we show how some of our results may be utilised in the context of coil designs for plasma fusion confinement devices in order to obtain coil configurations of particular simple shape.
Lastly, we consider the problem of optimising surface helicity among toroidal surfaces of fixed area and show that toroidal surfaces admitting a symmetry constitute global minimisers.37 pages, 5 figures. One citation has been corrected in comparison to version 1. The content of the new version remains otherwise unchange
Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes
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Typical field lines of Beltrami flows and boundary field line behaviour of Beltrami flows on simply connected, compact, smooth manifolds with boundary
No full textA unique continuation theorem for exterior differential forms on Riemannian manifolds with boundary
Get PDFAronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique
continuation result for differential forms, satisfying a certain first order
differential inequality, on Riemannian manifolds with empty boundary. The
present paper extends this result to the setting of Riemannian manifold with
non-empty boundary, assuming suitable boundary conditions on the differential
forms. We then present some applications of this extended result. Namely, we
show that the Hausdorff dimension of the zero set of harmonic Neumann and
Dirichlet forms, as well as eigenfields of the curl operator (on
-manifolds), has codimension at least . Again, these bounds were known in
the setting of manifolds without boundary, so that the merit is once more the
inclusion of boundary points.Comment: 9 pages, comments welcome
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