arXiv subjects

Wadim Gerner

Publications and source records attributed to Wadim Gerner.

At least 19 recordsLinked to original sources

arXiv2026

An analytic formula for surface currents generating prescribed plasma equilibrium fields

Given a plasma domain $P\subset\mathbb{R}^3$, a plasma equilibrium field $B$ on $P$ and a coil winding surface $\Sigma$ surrounding $P$, we provide an analytic formula whose output is a surface current distribution $j$ on $\Sigma$ such that $\operatorname{BS}(j)+\operatorname{BS}(\operatorname{curl}(B))=B$ in $P$, i.e. the combination of the plasma current magnetic field and the surface current magnetic field exactly produce the full plasma equilibrium field. We use this new formula to provide a theoretical explanation of the empirical phenomenon that currents tend to attain their maximal strength at points which correspond to points of highest normal magnetic curvature on the plasma boundary. Some discussions regarding aspects of numerical approximations are also included.

physics.plasm-ph↗

arXiv2026

Electric current dynamics in the stellarator coil winding surface model

In stellarator design, the coil winding surfaces $\Sigma\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.

physics.plasm-ph↗

arXiv2026

Inverse problems for quasi-linear elliptic systems modeling electrolysers

We investigate the electrochemical processes within an electrolyser cell, which are modelled by a coupled system of second-order quasi-linear elliptic PDEs. In this context, we study an inverse problem aiming to reconstruct both the non-linear diffusion coefficients and the phenomenological relation defining the electric potential. Our main results state that boundary measurements alone are not enough to reconstruct these non-linear quantities. However, we show that a combination of boundary and interior measurements allow for their unique reconstruction. To achieve this result we generalise a linearisation result in the context of the scalar quasi-linear Calder\'{o}n problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calder\'{o}n case, the generalised linearisation does not "freeze" the coefficients. We show that interior measurements are precisely what is required to achieve this freezing and thus enable the unique reconstruction.

math.AP↗

arXiv2026

Charged particle motion in a strong magnetic field: The first order expansion

We provide a mathematically rigorous derivation of the first order expansion of the motion of a charged particle in a strong magnetic field. In contrast to the derivations that can be found in the physics literature we solely assume throughout that the magnetic field is strong. In particular we do not need to make any structural assumptions on the particle motion, such as the gyroradius being small in comparison to the magnetic length scale. Instead, some of the additional assumptions which are usually made in the physics literature turn out to be an a posteriori consequence in our approach. Our approach further justifies the utilisation of the guiding centre approximation at "bounce points" within magnetic mirrors, a situation which violates the usual assumptions which are made in the physics literature when deriving the guiding centre approximation.

math-ph↗

arXiv2025

Non-optimal domains for the helicity maximisation problem

In [J. Cantarella, D. DeTurck, H. Gluck and M. Teytel, J. Math. Phys. 41:5615 (2000)] the helicity isoperimetric problem which asks to find a smooth domain of fixed volume which maximises Biot-Savart helicity among all other smooth domains of fixed volume was initiated. It was shown that if an optimal domain exists, all of its boundary components must be tori. The present work extends these results by establishing additional geometric constraints which optimal domains, if they exist, must satisfy. This allows to rule out the optimality of a broad class of solid tori. The existence of optimal domains remains an open problem.

math.AP↗

arXiv2025

Influence of coil geometry and coil-plasma distance on the magnetic field approximation error

We investigate analytically two questions: 1) How does the coil geometry influence the effect of electric current noise on the induced magnetic field? 2) How does the coil-plasma distance influence our ability to control the pointwise magnetic field error in terms of the average magnetic field error? Regarding (1), we argue that the main geometric quantities of interest are the notion of reach and the volume of the region enclosed by the coils. Our main finding is a quantitative formula which shows that the larger the reach and the smaller the volume of the region enclosed by the coils, the smaller is the influence of the electric current uncertainty on the magnetic fields. Regarding (2), we show that the pointwise magnetic field error can be controlled (modulo an explicit constant) by the average square-magnetic field-error times $(\text{coil-plasma distance})^{-\frac{3}{2}}$.

physics.plasm-ph↗

arXiv2025

Quantitative Gaffney and Korn inequalities

We prove a homogeneous, quantitative version of Ehrling's inequality for the function spaces $H^1(\Omega)\subset\subset L^2(\partial\Omega)$, $H^1(\Omega)\hookrightarrow L^2(\Omega)$ which reflects geometric properties of a given $C^{1,1}$-domain $\Omega\subset\mathbb{R}^n$. We use this result to derive quantitative homogeneous versions of Gaffney's inequality, of relevance in electromagnetism as well as Korn's inequality, of relevance in elasticity theory. The main difference to the corresponding classical results is that the constants appearing in our inequalities turn out to be dimensional constants. We provide explicit upper bounds for these constants and show that in the case of the tangential homogeneous Korn inequality our upper bound is asymptotically sharp as $n\rightarrow \infty$. Lastly, we raise the question of the optimal values of these dimensional constants.

math.AP↗

arXiv2025

Kernel and image of the Biot-Savart operator and their applications in stellarator designs

We consider the Biot-Savart operator acting on $W^{-\frac{1}{2},2}$ regular, div-free, surface currents $j$ \begin{gather} \nonumber \operatorname{BS}(j)(x)=\frac{1}{4\pi}\int_{\Sigma}j(y)\times \frac{x-y}{|x-y|^3}d\sigma(y)\text{, }x\in \Omega \end{gather} where $\Sigma$ is a connected surface to which $j$ is tangent and where $\Omega$ is the finite domain bounded by $\Sigma$. We answer two questions regarding this operator. i) We provide an algorithm which converges (theoretically) exponentially fast to an element of the kernel of the Biot-Savart operator, as well as characterise the elements of the kernel of the Biot-Savart operator in terms of certain solutions to exterior boundary value problems. This allows one to explicitly exploit the non-uniqueness of the coil reconstruction process in the context of stellarator designs. ii) We provide a simple, concise characterisation of the image of the Biot-Savart operator. This allows to define a 2-step current reconstruction procedure to obtain surface currents which approximate to arbitrary precision a prescribed target magnetic field within the plasma region of a stellarator device. The first step does not require computing integrals involving singular integral kernels of the form $\frac{x-y}{|x-y|^3}$ but may have a potentially slow convergence rate, while the second step requires the computation of integrals involving singular integral kernels but in turn has (theoretically) an exponential convergence rate. This approximation procedure always leads to approximating surface currents which are as poloidal as possible.

math-ph↗

arXiv2025

Charged particle motion in a strong magnetic field: Applications to plasma confinement

We derive the zero order approximation of a charged particle under the influence of a strong magnetic field in a mathematically rigorous manner and clarify in which sense this approximation is valid. We use this to further rigorously derive a displacement formula for the pressure of plasma equilibria and compare our findings to results in the physics literature. The main novelty of our results is a qualitative estimate of the confinement time for optimised plasma equilibria with respect to the gyro frequency. These results are of interest in the context of plasma fusion confinement.

math-ph↗

arXiv2024

Asymptotic windings, surface helicity and their applications in plasma physics

In [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.

math.DS↗

arXiv2023

Properties of the Biot-Savart operator acting on surface currents

We investigate properties of the image and kernel of the Biot-Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot-Savart operator is $L^2$-dense in the space of square-integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighbourhood of the underlying plasma domain can in fact be approximated in any $C^k$-norm by elements of the image of the Biot-Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot-Savart operator which in particular implies that the dimension of the kernel of the Biot-Savart operator coincides with the genus of the coil winding surface and hence turns out to be a homotopy invariant among regular domains in $3$-space. Lastly, we provide an iterative scheme which we show converges weakly in $W^{-\frac{1}{2},2}$-topology to elements of the kernel of the Biot-Savart operator.

math-ph↗

arXiv2023

Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition

In the present work we present a general framework which guarantees the existence of optimal domains for isoperimetric problems within the class of $C^{1,1}$-regular domains satisfying a uniform ball condition as long as the desired objective function satisfies certain properties. We then verify that the helicity isoperimetric problem studied in [J. Cantarella, D. DeTurck, H. Gluck and M. Teytel, J. Math. Phys. 41, 5615 (2000)] satisfies the conditions of our framework and hence establish the existence of optimal domains within the given class of domains. We additionally use the same framework to prove the existence of optimal domains among uniform $C^{1,1}$-domains for a first curl eigenvalue problem which has been studied recently for other classes of domains in [A. Enciso, W. Gerner and D. Peralta-Salas, Trans. Amer. Math. Soc. 377, 4519-4540 (2024)].

math-ph↗

arXiv2023

Optimal metrics for the first curl eigenvalue on 3-manifolds

In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the $L^2$-norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that $\mathbf{S}^3$ and $\mathbf{R}P^3$ endowed with the round metric are local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of $\mathbf{S}^3$ and $\mathbf{R}P^3$ are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in dimension 4.

math.DG↗

arXiv2022

A unique continuation theorem for exterior differential forms on Riemannian manifolds with boundary

Aronszajn, 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 $3$-manifolds), has codimension at least $2$. Again, these bounds were known in the setting of manifolds without boundary, so that the merit is once more the inclusion of boundary points.

math.AP↗

arXiv2022

Isoperimetric problem for the first curl eigenvalue

We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by getting rid of the connected boundary assumption. Lastly, we show how a second variation inequality related to the isoperimetric problem may be used in order to relate the existence of Killing-Beltrami fields to the geometry of the ambient space.

math.AP↗

arXiv2022

Optimal convex domains for the first curl eigenvalue

We prove that there exists a bounded convex domain $\Omega \subset \mathbf{R}^3$ of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class $C^{1,1}$). Existence results for uniformly H\"older optimal domains in a box (that is, contained in a fixed bounded domain $D \subset \mathbf{R}^3$) are also presented.

math.AP↗

arXiv2020

Existence and structure of symmetric Beltrami flows on compact $3$-manifolds

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $\phi_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field $X$, tangent to the boundary, which is an eigenfield of curl and satisfies $(\phi_t)_{*}X=X$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\bar{M},g)$ has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable $3$-manifolds with and without boundary.

math.AP↗

arXiv2020

Zero set structure of real analytic Beltrami fields

In this paper we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected $3$-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected $1$-dimensional submanifolds with (possibly empty) boundary and tame knots. Further we consider the question of how complicated these tame knots can possibly be. To this end we prove that on the standard (open) solid toroidal annulus in $\mathbb{R}^3$, there exist for any pair $(p,q)$ of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric there exists a real analytic Beltrami field, corresponding to the eigenvalue $+1$ of the curl operator, whose zero set is precisely given by a standard $(p,q)$-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

math.DS↗