arXiv subjects

David Perrella

Publications and source records attributed to David Perrella.

8 recordsLinked to original sources

arXiv2025

Asymmetry of curl eigenfields solving Woltjer's variational problem

We construct families of rotationally symmetric toroidal domains in $\mathbb R^3$ for which the eigenfields associated to the first (positive) Amp\`erian curl eigenvalue are symmetric, and others for which no first eigenfield is symmetric. This implies, in particular, that minimizers of the celebrated Woltjer's variational principle do not need to inherit the rotational symmetry of the domain. This disproves the folk wisdom that the eigenfields corresponding to the lowest curl eigenvalue must be symmetric if the domain is.

math-ph↗

arXiv2024

Closed orbits of MHD equilibria with orientation-reversing symmetry

As a generalisation of the periodic orbit structure often seen in reflection or mirror symmetric MHD equilibria, we consider equilibria with other orientation-reversing symmetries. An example of such a symmetry, which is a not a reflection, is the parity transformation $(x,y,z) \mapsto (-x,-y,-z)$ in $\mathbb{R}^3$. It is shown under any orientation-reversing isometry, that if the pressure function is assumed to have toroidally nested level sets, then all orbits on the tori are necessarily periodic. The techniques involved are almost entirely topological in nature and give rise to a handy index describing how a diffeomorphism of $\mathbb{R}^3$ alters the poloidal and toroidal curves of an invariant embedded 2-torus.

math.DS↗

arXiv2024

Global realisation of magnetic fields as 1$\frac{1}{2}$D Hamiltonian systems

The paper reviews the notion of $n+\frac{1}{2}$D non-autonomous Hamiltonian systems, portraying their dynamics as the flow of the Reeb field related to a closed two-form of maximal rank on a cosymplectic manifold, and naturally decomposing into time-like and Hamiltonian components. The paper then investigates the conditions under which the field-line dynamics of a (tangential) divergence-free vector field on a connected compact three-manifold (possibly with boundary) diffeomorphic to a trivial fibre bundle over the circle can be conversely identified as a non-autonomous $1\frac{1}{2}$D Hamiltonian system. Under the assumption that the field is transverse to a global compact Poincar\'e section, an adaptation of Moser's trick shows that all such fields are locally-Hamiltonian. A full identification is established upon further assuming that the Poincar\'e sections are planar, which crucially implies (together with Dirichlet boundary conditions) that the cohomology class of the generating closed one-forms on each section is constant. By reviewing the classification of fibre bundles over the circle using the monodromy representation, it is remarked that as soon as the Poincar\'e section of an alleged field is diffeomorphic to a disk or an annulus, the domain is necessarily diffeomorphic to a solid or hollow torus, and thus its field-line dynamics can always be identified as a non-autonomous Hamiltonian system.

math-ph↗

arXiv2023

Asymmetry of MHD equilibria for generic adapted metrics

Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution $X$ admits a large set of ``adapted" metrics in $M$ for which $X$ solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in $\mathbb{R}^3$ with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.

math.DG↗

arXiv2023

Existence of global symmetries of divergence-free fields with first integrals

The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type Theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type Theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality; without needing to assume a symmetry such as in the cases of magneto-hydrostatics (MHS) or quasi-symmetry.

math.DG↗

arXiv2022

Linearisability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field $B$ to an invariant toroidal surface $S$ is linearisable. The main results are similar in conclusion to Arnold's Structure Theorems but require weaker assumptions than the commutation $[B,\nabla\times B] = 0$. Relaxing the need for a first integral of $B$ (also known as a flux function), we assume the existence of a solution $u : S \to \mathbb{R}$ to the cohomological equation $B|_S(u) = \partial_n B$ on a toroidal surface $S$ mutually invariant to $B$ and $\nabla \times B$. The right hand side $\partial_n B$ is a normal surface derivative available to vector fields tangent to $S$. In this situation, we show that the field $B$ on $S$ is either identically zero or nowhere vanishing with $B|_S/\|B\|^2 |_S$ being linearisable. We are calling the latter the semi-linearisability of $B$ (with proportionality $\|B\|^2 |_S$). The non-vanishing property relies on Bers' results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that $B|_S$ itself is linearisable. The linearisability of $B|_S$ is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

math.DG↗

arXiv2021

Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary

This paper completes the foundations of neatly integrable normal distribution theory on manifolds with boundary. Normal distributions are those which contain vectors transverse to the boundary along its entirety. The theory is observed to be entirely analogous with the theory of integrable distributions on manifolds due to Stefan and Sussmann. The main result is a one-to-one correspondence between so-called neatly integrable normal distributions and neat foliations by manifolds with boundary. Neat foliations are allowed to have non-constant dimension and the leaves have boundary contained in the ambient boundary. The leaves satisfy a characteristic property formally identical to that of weakly embedded submanifolds except in the category of manifolds with boundary.

math.DG↗

arXiv2021

A Stefan-Sussmann theorem for normal distributions on manifolds with boundary

An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to "integrate" a normal distribution; the next best "integrals" are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.

math.DG↗