Linear Magnetohydrodynamic Waves in a Magneto-Lattice: A Unified Theoretical Framework and Numerical Validation

TL;DR

The paper tackles how linear MHD waves propagate in a spatially periodic magnetic background, a magneto-lattice. It develops two equivalent central equations via Bloch's theorem and the plane wave expansion, one in $( ho,oldsymbol{B},oldsymbol{v})$ and one in displacement $oldsymbol{\xi}$, and demonstrates their equivalence by analytical and numerical means. Using an isothermal, periodic background, the authors show that periodicity creates intrinsic frequency bandgaps, broadening with the modulation amplitude $B_m$, and that Alfvén waves split into multiple branches not present in uniform plasmas. Validation comes from truncated central equations and full MHD simulations (Athena++), which confirm bandgap locations and widths and the AW splitting, highlighting a pathway to tunably control MHD wave propagation in structured plasmas and laying groundwork for higher-dimensional magneto-lattice explorations.

Abstract

We present a systematic theoretical and numerical investigation of the propagation properties of linear magnetohydrodynamic (MHD) waves in a spatially periodic magnetic field, referred to as a magneto-lattice. Two types of central equations, expressed in terms of $\left(ρ,\boldsymbol{B},\boldsymbol{v}\right)$ (where $ρ$ is perturbed mass density, $\boldsymbol{B}$ is perturbed magnetic field, and $\boldsymbol{v}$ is perturbed velocity) and the perturbation displacement $\boldsymbolξ$, are established using the plane wave expansion (PWE) method. The validity of both equations is demonstrated through two numerical examples. This framework enables the identification of intrinsic frequency bandgaps and cutoff phenomena within the system. Our numerical results show that the bandgap width increases with the magnetic modulation ratio $B_{m}$, leading to the suppression of specific MHD wave modes. Furthermore, the periodicity of the magnetic field induces the splitting of Alfvén waves into multiple branches\textemdash a phenomenon absent in uniform plasmas. These findings provide new insights for manipulating MHD waves in a crystalline lattice framework of structured plasmas.

Figures (4)

• Figure 1: Schematic illustration of periodic lattice structures in different physical systems. (a) Atomic crystal lattice in condensed matter physics. (b) Periodic elastic lattice in a phononic crystal for manipulating acoustic or elastic waves. (c) Periodic dielectric lattice in a photonic crystal for controlling electromagnetic wave propagation. (d) Magneto-lattice formed by periodically arranged magnetic dipoles, producing a spatially modulated magnetic field for controlling the propagation of MHD waves.
• Figure 2: Band-structure (dispersion-relation) benchmark for two central equations with $k_{y}=1$ and $k_{z}=0$. (a) Dispersion relations at $B_{m}=0$. Hollow circles and solid dots denote results obtained from the $\left(\rho,\boldsymbol{B},\boldsymbol{v}\right)$, formulation and the $\boldsymbol{\xi}$ formulation, respectively. (b) Maximum frequency difference $\Delta\omega$ between the two dispersion relations as a function of $k_{x}$ at $B_{m}=0$. (c) Same as (a), but for $B_{m}=0.1$. (d) Same as (b), but for $B_{m}=0.1$.
• Figure 3: Comparison of results from three models under the empty lattice approximation. The solid curves represent the analytical solution, the discrete points represent the results from the truncated central equations, and the heatmap represents the background power spectrum of the full MHD simulation performed with the Athena++ code. The dispersion relations for fast waves, slow waves, and Alfvén waves were calculated individually and subsequently folded into the first Brillouin zone.
• Figure 4: Comparison between Athena++ simulations and the truncated central equation. The solid line represents the result from the truncated central equation, while the power spectrum shows the numerical result from the Athena++ simulation. (a) fast wave with $B_{m}=0.1$, (b) Alfvén wave with $B_{m}=0.1$, (c) fast wave with $B_{m}=0.2$, (d) Alfvén wave with $B_{m}=0.2$.