Simons Hour Talk

Spatial Dynamics Formulation of Magnetohydrostatics
Feb 29, 2024, 10:00 am11:00 am


Event Description

Abstract: The magnetohydrostatics (MHS) PDE system is usually studied as a 3-dimensional boundary value problem. I will describe several benefits to treating MHS as a 2-dimensional ​"time" evolution problem, where toroidal angle plays the role of "time" and poloidal coordinates parameterize "space". In this spatial dynamics reformulation of MHS, planar hydrodynamic equations prescribe the evolution of toroidal flux, pressure, and field line velocity. Newcomb's variational principle for the usual boundary-value formulation of MHS implies a dynamical variational principle for the hydrodynamic system. I will summarize the implications of this variational principle in light of Noether's theorem connecting symmetries with conservation laws. As with all variational dynamical systems, the MHS spatial dynamics equations admit a dual Hamiltonian formulation. Remarkably, this Hamiltonian structure is Lie-Poisson on the dual to a certain infinite-dimensional Lie algebra. This implies that a technique developed by Scovel-Weinstein originally for beam physics produces exact particle-swarm solutions of the spatial dynamics equations suitably regularized at small poloidal length scales, thereby suggesting a method for sidestepping Grad's conjecture. The particle-like solutions obey an ODE system that I call smoothed particle magnetohydrostatics (SPMHS), due to its resemblance with smoothed particle hydrodynamics. In large aspect ratio domains, the spatial dynamics equations exhibit a pair of disparate timescales. Reducing to the corresponding normally-hyperbolic slow manifold corresponds to solving the elliptic part of the equations, leaving a hyperbolic evolution equation that every physical solution must obey. Eliminating the elliptic component of MHS in this manner eliminates the confounding mixed type of the original PDE system, thereby opening the door to applying (Hamiltonian) hyperbolic PDE theory to stellarator equilibria. 

Talk time in other timezones:
AEDT  2:00 AM Fri  1 Mar,
JST  12:00 AM Fri  1 Mar,
CET   4:00 PM Thu 29 Feb,
GMT   3:00 PM Thu 29 Feb,
UTC  15:00    Thu 29 Feb,
EST  10:00 AM Thu 29 Feb,
CST   9:00 AM Thu 29 Feb,
MST   8:00 AM Thu 29 Feb,
PST   7:00 AM Thu 29 Feb
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