arXiv subjects

Michael Roop

Publications and source records attributed to Michael Roop.

4 recordsLinked to original sources

arXiv2026

Hamiltonian formulation and matrix discretization for axisymmetric magnetohydrodynamics

Equations of ideal magnetohydrodynamics (MHD) play an important role in the studies of turbulence, astrophysics, and plasma physics. These equations possess remarkable geometric structures and symmetries. Indeed, they admit a geodesic formulation in the sense of Arnold, as a Lie--Poisson flow on the dual of an infinite-dimensional Lie algebra. Zeitlin's model, previously developed for MHD on the flat torus and the two-sphere, is a matrix approximation of MHD consistent with the underlying geometric structures. In this paper, we derive the reduced model of axially symmetric magnetohydrodynamics on the three-sphere and give its Hamiltonian formulation. We further extend finite dimensional Zeitlin's matrix model for MHD from 2D to axially symmetric 3D flows of magnetized fluids, yielding the first discrete model for 3D magnetohydrodynamics compatible with the underlying Lie--Poisson structure.

math-ph↗

arXiv2025

Structure-preserving long-time simulations of turbulence in magnetized ideal fluids

We address three two-dimensional magnetohydrodynamics models: reduced magnetohydrodynamics (RMHD), Hazeltine's model, and the Charney--Hasegawa--Mima (CHM) equation. These models are derived to capture the basic features of magnetohydrodynamic turbulence and plasma behaviour. They all possess non-canonical Hamiltonian formulations in terms of Lie--Poisson brackets, which imply an infinite number of conservation laws along with symplecticity of the phase flow. This geometric structure in phase space affects the statistical long-time behaviour. Therefore, to capture the qualitative features in long-time numerical simulations, it is critical to use a discretization that preserves the rich phase space geometry. Here, we use the matrix hydrodynamics approach to achieve structure-preserving discretizations for each model. We furthermore carry out long-time simulations with randomized initial data and a comparison between the models. The study shows consistent behaviour for the magnetic potential: both RMHD and Hazeltine's model produce magnetic dipoles (in CHM, the magnetic potential is prescribed). These results suggest an inverse cascade of magnetic energy and of the mean-square magnetic potential, which is empirically verified via spectral scaling diagrams. On the other hand, the vorticity field dynamics differs between the models: RMHD forms sharp vortex filaments with rapidly growing vorticity values, whereas Hazeltine's model and CHM show only small variation in the vorticity values. Related to this observation, both Hazeltine's model and CHM give spectral scaling diagrams indicating an inverse cascade of kinetic energy not present in RMHD.

physics.plasm-ph↗

arXiv2025

Thermal quasi-geostrophic model on the sphere: derivation and structure-preserving simulation

We derive the global model of thermal quasi-geostrophy on the sphere via asymptotic expansion of the thermal rotating shallow water equations. The model does not rely on the asymptotic expansion of the Coriolis force and extends the quasi-geostrophic model on the sphere by including an additional transported buoyancy field acting as a source term for the potential vorticity. We give its Hamiltonian description in terms of semidirect product Lie--Poisson brackets. The Hamiltonian formulation reveals the existence of an infinite number of conservation laws, Casimirs, parameterized by two arbitrary smooth functions. A structure-preserving discretization is provided based on Zeitlin's self-consistent matrix approximation for hydrodynamics. A Casimir-preserving time integrator is employed to numerically fully preserve the resulting finite-dimensional Lie--Poisson structure. Simulations reveal the formation of vorticity and buoyancy fronts, and large-scale structures in the buoyancy dynamics induced by the buoyancy-bathymetry interaction.

math.NA↗

arXiv2023

Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere

We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semi-direct product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie--Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model.

math.NA↗