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Register a new userOn 2D Harmonic Extensions of Vector Fields and Stellarator Coils
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We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned. Current-carrying coils are represented as singularities within the magnetic field and therefore this problem can be modelled mathematically as finding how far we can harmonically extend a vector field from the boundary of a domain. For this paper we consider two-dimensional domains with real analytic boundary, and prove that a harmonic extension exists if and only if the boundary data satisfies a combined compatibility and regularity condition. Our method of proof uses a generalisation of a result of Hadamard on the Cauchy problem for the Laplacian. We then provide a lower bound on how far we can harmonically extend the vector field from the boundary via the Cauchy--Kovalevskaya Theorem.
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Magnetic confinement devices for nuclear fusion can be large and expensive. Compact stellarators are promising candidates for costreduction, but introduce new difficulties: confinement in smaller volumes requires higher magnetic field, which calls for higher coil-currents and ultimately causes higher Laplace forces on the coils-if everything else remains the same. This motivates the inclusion of force reduction in stellarator coil optimization. In the present paper we consider a coil winding surface, we prove that there is a natural and rigorous way to define the Laplace force (despite the magnetic field discontinuity across the current-sheet), we provide examples of cost associated (peak force, surface-integral of the force squared) and discuss easy generalizations to parallel and normal force-components, as these will be subject to different engineering constraints. Such costs can then be easily added to the figure of merit in any multi-objective stellarator coil optimization code. We demonstrate this for a generalization of the REGCOIL code [1], which we rewrote in python, and provide numerical examples for the NCSX (now QUASAR) design. We present results for various definitions of the cost function, including peak force reductions by up to 40 %, and outline future work for further reduction.
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.
A neoclassically optimized compact stellarator with simple coils has been designed. The magnetic field of the new stellarator is generated by only four planar coils including two interlocking coils of elliptical shape and two circular poloidal field coils. The interlocking coil topology is the same as that of the Columbia Non-neutral Torus (CNT). The new configuration was obtained by minimizing the effective helical ripple directly via the shape of the two interlocking coils. The optimized compact stellarator has very low effective ripple in the plasma core implying excellent neoclassical confinement. This is confirmed by the results of the drift-kinetic code SFINCS showing that the particle diffusion coefficient of the new configuration is one order of magnitude lower than CNTs.
In this paper we consider the gradient flow of the following Ginzburg-Landau type energy [ F_varepsilon(u) := frac{1}{2}int_{M}vert D uvert_g^2 +frac{1}{2varepsilon^2}left(vert uvert_g^2-1right)^2mathrm{vol}_g. ] This energy is defined on tangent vector fields on a $2$-dimensional closed and oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative) and depends on a small parameter $varepsilon>0$. If the energy satisfies proper bounds, when $varepsilonto 0$ the second term forces the vector fields to have unit length. However, due to the incompatibility for vector fields on $M$ between the Sobolev regularity and the unit norm constraint, critical points of $F_varepsilon$ tend to generate a finite number of singular points (called vortices) having non-zero index (when the Euler characteristic is non-zero). These types of problems have been extensively analyzed in a recent paper by R. Ignat and R. Jerrard. As in Euclidean case, the position of the vortices is ruled by the so-called renormalized energy. In this paper we are interested in the dynamics of vortices. We rigorously prove that the vortices move according to the gradient flow of the renormalized energy, which is the limit behavior when $varepsilonto 0$ of the gradient flow of the Ginzburg-Landau energy.
We derive a lower bound for energies of harmonic maps of convex polyhedra in $ R^3 $ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.
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