Mini Lecture course
Daniel Peralta-Salas (ICMAT)
Lecture 1: Magnetohydrostatic equilibria in toroidal geometries (Video)
The computation of 3D magnetohydrostatic (MHS) equilibria is of major importance for magnetic confinement devices such as tokamaks or stellarators. In this talk I will present recent results on the existence of stepped pressure MHS equilibria in 3D toroidal domains, where the plasma current exhibits an arbitrary number of current sheets. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The proof involves three main ingredients: a Cauchy-Kovalevskaya theorem for Beltrami fields, a Hamilton-Jacobi equation on the two-dimensional torus, and a KAM theorem for 3D solenoidal fields. This is based on joint work with A. Enciso and A. Luque.
Lecture 2: Obstructions to topological relaxation for generic magnetic fields (Video)
Magnetic relaxation is a mechanism that aims to obtain magnetohydrostatic (MHS) equilibria as long-time limits of the ideal magnetohydrodynamics equations. My goal in this talk is to present generic obstructions for a volume preserving vector field to be topologically equivalent to some MHS equilibrium. Specifically, I will show that for any axisymmetric toroidal domain there is a locally generic set of volume preserving fields that are not topologically equivalent to any MHS equilibrium. Each vector field in this set is Morse-Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. As a main motivation, I will explain how this result is related to the celebrated Parker's hypothesis on plasmas in stellar atmospheres, which is wide open since 1972. This is based on joint work with A. Enciso.
March 26th
David Pfefferlé (The University of Western Australia)
Pieces of the 3D MHS puzzle (Slide, Video)
We will explore various peculiarities of the magneto-statics problems, the necessary conditions for equilibrium, the irritating obstructions to solutions and the numerical evidence of the mathematical challenges in finding them.
David Perrella (The University of Western Australia)
Topology alone is too soft to answer the Grad Conjecture (Slide, Video)
We focus on the question of whether purely topological methods (at the level of the space of metric spaces) can resolve the Grad Conjecture. Arnold's Structure Theorem and symmetires apply to MHD equilibria but do not completely characterise them. We find that a fixed MHD equilibrium 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.
Yasuhide Fukumoto (Kyushu University)
Particle-relabeling as variational and divergence symmetries for ideal fluid dynamics and magnetohydrodynamics (Slide, Video)
The helicity is a topological invariant of ideal Euler flows in three dimensions. This is no longer the case if the baroclinic effect and, for a conducting fluid, the Lorentz force are called into play. By appealing to Noether's theorem, we show that the cross-helicity is the integral invariant associated with the particle-relabeling symmetry of the action for ideal magnetohydrodynamics. Employing the Lagrange label function, as the independent variable in the variational framework, facilitates implementation of the relabeling transformation. The Casimirs in the form of integrals including Lagrangian invariants like the Ertel invariant are found to be variants of the cross-helicity. By incorporating the divergence symmetry, other known topological invariants are put on the same ground of Noether's theorem.
Yukimi Goto (Kyushu University)
Phase Transition in a Lattice Nambu–Jona-Lasinio Model (Slide, Video)
In this talk I will discuss a phase transition in a lattice model of QCD called the Nambu-Jona-Lasinio model. The four-fermion interaction has been widely used as an effective model to describe the low-energy behavior of strongly interacting quarks. This talk is based on work (Commun. Math. Phys. 2023; arXiv:2310.15922) with Tohru Koma.
Ikkei Shimizu (Osaka University)
On mathematical study of the Landau-Lifshitz energy with helicity term (Slide, Video)
The Landau-Lifshitz energy is one of the phenomenological mathematical model to describe the magnetization of ferromagnets. Recently, the energy with helicity term, known as the Dzyaloshinskii-Moriya interaction, is paid increasing attention in physics, since it can explain the formation of skyrmion and various kind of patterns. In this talk, we overview the mathematical study on this energy, such as variational problems, Cauchy problems for corresponding dynamical PDEs, etc.
March 27th
Javier Peñafiel-Tomás (ICMAT)
Extension of Euler flows (Slide, Video)
Given a smooth solution of the Euler equations in a bounded domain, we show how to extend it to a global weak solution. We use an adapted convex integration scheme as well as new results concerning the construction, extension and gluing of subsolutions. These techniques yield high spatial control on the subsolutions throughout the iterative process. Time permitting, I will discuss some other applications of these ideas.
Koji Ohkitani (Kyoto University)
Diffusive analogues of first integrals for incompressible fluid dynamics and magnetohydrodynamics: a survey (Slide, Video)
After recalling Cauchy formula for the 3D Euler equations and Lundquist formula for magnetohydrodynamics, we review their extensions to diffusive dynamical equations.
Kiori Obuse (Okayama University)
Rossby wave nonlinear interactions and large-scale zonal flow formation in two-dimensional turbulence on a rotating sphere (Slide, Video)
Two-dimensional barotropic flow on a rotating sphere is one of the simplest mathematical models describing the dynamics of planetary atmospheres. The model is very simple and does not take into account, for example, three-dimensional fluid motion, planetary topography and heat distribution. Nevertheless, it exhibits rich fluid dynamics, including the formation of large-scale zonal flows [1,2,3,4]. This model also has the interesting aspect that it has the same mathematical structure as the special case of Hasegawa-Mima equation, which is often used in plasma physics, when a plane approximation is applied to it.
In this talk, we consider unforced two-dimensional turbulence on a rotating sphere and discuss how the nonlinear interactions of Rossby waves, which are linear wave solutions unique to rotating systems, are involved in the formation of the large-scale zonal flows (westward circumpolar flows in this case.). It is known that when the rotation rate of the sphere is very high, the three-wave resonance non-linear interaction of Rossby wave strongly dominates the dynamics of the flow field [5,6]. However, it is not possible to transfer energy to Rossby waves corresponding to zonal flows (zonal Rossby waves) directly by three-wave resonance interactions [7,8]. This means that the formation of zonal flows takes place by weakly existing non-resonant interactions, but the details have been little understood. In particular, we still do not know why the zonal flows that develop due to non-resonant interactions consist of waves that are capable of resonant interactions, rather than waves that are incapable of non-resonant interactions [8].
Based on our recent detailed numerical calculations of energy transfer by Rossby wave three-wave non-resonant interactions, we report that the formation of the westward circumpolar flow is due to non-local energy transfer by three-wave near-resonant interactions (special cases of non-resonant interactions) .
References
[1] S. Yoden and M. Yamada, J. Atomos. Sci. 50, 631 (1993)
[2] S. Takehiro et al., J. Atmos. Sci. 64, 4084 (2007)
[3] T. Nozawa and S. Yoden, Phys. Fluids 9, 2081 (1997)
[4] K. Obuse et al., Phys. Fluids. 22, 056601 (2010)
[5] M. Yamada and T. Yoneda, Physica D 245, 1 (2013)
[6] A. Dutrifoy and M. Yamada, In preparation.
[7] G. M. Reznik et al., Dyn. Atmos. Oceans 18, 235 (1993).
[8] K. Obuse and M. Yamada, Phys. Rev. Fluids 4, 024601 (2019)
Tomoo Yokoyama (Saitama University)
Topological study of 2D flows and its application (Slide, Video)
We introduce a topological method for representing generic fluid phenomena in two-dimensional space without ambiguity. In other words, a framework, called topological flow data analysis, to study topological properties of flows is introduced. First, we apply these methods to flows in the human body, ocean currents, flow around airfoils, flow in machines, and so on. Second, the existing dynamical system theory is overviewed, and generic 2D Hamiltonian flows and their generic transitions are described. Furthermore, we introduce topological representations of gradient flows and incompressible flows in a unified manner to construct a foundation for analyzing fluid phenomena.
Takeda Keiichiro (The University of Tokyo)
Clebsch representation and generalized enstrophy for relativistic plasma (Slide, Video)
The theory of relativistic plasmas is useful to model high-energy astronomical objects. In this context, topological constraints built in the governing equations play an essential role in characterizing structures self-organized by plasmas. Among various invariants of ideal models, the circulation is one of the most fundamental quantities, being included in other invariants like the helicity. The conventional enstrophy, known to be constant in a two-dimensional flow, can be generalized, by invoking Clebsch variables, to the topological charge of a three-dimensional fluid element, which essentially measures circulations.
Since the relativistic effect imparts space-time coupling into the metric, such invariants must be modified. The non-relativistic generalized enstrophy is no longer conserved in a relativistic plasma, implying that the conservation of circulation is violated. In this work, we extend the generalized enstrophy to a Lorentz covariant form. We formulate the Clebsch representation in relativity using the principle of least action and derive a relativistically modified generalized enstrophy that is conserved in the relativistic model. We also calculate toy models and show that relativistic effects cannot be apparent in highly symmetric flows.
March 28th
Federico Pasqualotto (UC Berkeley)
On the construction of MHD equilibria in general bounded domains (Slide, Video)
In this talk, I will present a construction of MHD equilibria on a general 3D bounded domain. Our construction is achieved by considering a regularized MHD system and showing that it gives rise to an MHD equilibrium in the infinite time limit. The regularization we impose on the MHD system is inviscid, and our procedure constitutes a rigorous justification of “magnetic relaxation” in this setup. This is joint work with Peter Constantin.
Ken Abe (Osaka Metropolitan University)
Existence of homogeneous Euler flows of degree −α ∉ [−2, 0] (Slide)
In this talk, I will discuss dilation invariant (homogeneous) solutions to the three-dimensional stationary Euler equations. Shvydkoy (2018) demonstrated the nonexistence of (−α)-homogeneous solutions in R³ \ {0} in the range 0 ≤ α ≤ 2 for the Beltrami and axisymmetric flows. I will discuss the existence of axisymmetric (−α)-homogeneous solutions in the complementary range α ∈ R \ [0, 2].
Naoki Sato (NIFS)
A Reduced Ideal MHD System for Nonlinear Magnetic Field Turbulence in Plasmas with Approximate Flux Surfaces (Slide, Video)
This work studies the nonlinear evolution of magnetic field turbulence in proximity of steady ideal MHD configurations characterized by a small electric current, a small plasma flow, and approximate flux surfaces, a physical setting that is relevant for plasma confinement in stellarators. The aim is to gather insight on magnetic field dynamics, to elucidate accessibility and stability of three-dimensional MHD equilibria, as well as to formulate practical methods to compute them. Starting from the ideal MHD equations, a reduced dynamical system of two coupled nonlinear PDEs for the flux function and the angle variable associated with the Clebsch representation of the magnetic field is obtained. It is shown that under suitable boundary and gauge conditions such reduced system preserves magnetic energy, magnetic helicity, and total magnetic flux. The noncanonical Hamiltonian structure of the reduced system is identified, and used to show the nonlinear stability of steady solutions against perturbations involving only one Clebsch potential. The Hamiltonian structure is also applied to construct a dissipative dynamical system through the method of double brackets. This dissipative system enables the computation of MHD equilibria by minimizing energy until a critical point of the Hamiltonian is reached. Finally, an iterative scheme based on the alternate solution of the two steady equations in the reduced system is proposed as a further method to compute MHD equilibria. A theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium as long as solutions exist at each step of the iteration. This is based on joint work arXiv:2311.03095 with M. Yamada.