On the double Beltrami states in Hall magnetohydrodynamics
Hantaek Bae, Kyungkeun Kang, Jaeyong Shin
TL;DR
This work analyzes double Beltrami states in Hall MHD by formulating them as steady solutions with the structure $B+ abla imes u=\alpha u$ and $u- abla imes B=-eta B$, and further classifies these states as sums of Beltrami flows under a constant $ abla(\alpha-eta)$. A variational principle using two helicities, $ ext{H}_B$ and $ ext{H}_{B+ abla imes u}$, is developed to obtain the double Beltrami form as energy-minimizing configurations, and conserved quantities are established under suitable function-space conditions. The time-dependent Hall MHD problem with equal viscosity and resistivity admits explicit exponential-decay solutions for non-degenerate cases and a degenerate form when necessary, with stability results showing global well-posedness and decay of perturbations. The analysis employs Littlewood–Paley techniques and Besov/Gagliardo–Nirenberg-type estimates to prove conservation laws and to derive global, perturbed stability and decay theorems. Together, these results advance understanding of the structure, dynamics, and stability of double Beltrami states in Hall MHD and provide rigorous foundations for their long-time behavior.
Abstract
In this paper, we investigate double Beltrami states in the Hall magnetohydrodynamic (Hall MHD) equations. Initially, we examine the double Beltrami states as a special class of steady solutions to the ideal Hall MHD equations, which are closely related to Beltrami flows in incompressible fluid dynamics. Specifically, we classify the double Beltrami states and show that they can be derived by using the variational method as energy minimizers, subject to the conservation of two helicities. We then extend our analysis to time-dependent double Beltrami states in the viscous and resistive Hall MHD equations, exploring their exact form and stability properties.