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

TL;DR

The paper addresses long-time turbulence in three 2D reduced MHD models (RMHD, Hazeltine, CHM) by employing structure-preserving matrix discretizations (Zeitlin) that implement Lie--Poisson geometry on finite-dimensional $ ext{su}(N)$ algebras and exact Casimir conservation. The authors derive RMHD--Zeitlin, Hazeltine--Zeitlin, and CHM--Zeitlin equations, using a Hoppe–Yau Laplacian and a geometric time integrator that preserves invariants and converges to the continuous system as $N oty$. Long-time simulations with randomized initial data reveal distinct dynamical behaviors: RMHD shows a forward kinetic-energy cascade with vorticity amplification and an inverse cascade of magnetic energy and the mean-square magnetic potential; Hazeltine exhibits an inverse cascade of kinetic energy and large-scale vortex blobs with a robust magnetic dipole, while CHM forms four vortex blobs with characteristic spectral slopes. The results underscore the importance of preserving Lie--Poisson structure for reliable long-time statistics and offer a framework for comparing reduced MHD models on the sphere, with implications for understanding magnetic turbulence and energy transfer mechanisms.

Abstract

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.

Figures (14)

• Figure 1: Overview of the relation between reduced models of magnetohydrodynamics.
• Figure 2: RMHD: Evolution of the velocity stream function $\psi(t)$ (left) and the magnetic potential $\theta(t)$ (right). The magnetic potential $\theta$ develops into the dipole configuration through intermediate mixing. The stream function $\psi$ does not develop large-scale structures, but one can observe circulations at locations of the magnetic eddies.
• Figure 3: RMHD: Evolution of the vorticity field supremum norm $\lVert \omega\rVert_\infty$. (a) For a simulation with spatial resolution $N=512$, the value initially grows rapidly and then reach a plateau at about $t=100$. (b) For simulations with the same initial data, the plateau is larger in magnitude for higher spatial resolution. This indicates that the value grows indefinitely as $N\to \infty$.
• Figure 4: RMHD: Evolution of the vorticity $\omega(t)$ (left) and the current density $j(t)$ (right). Vorticity and current density islands resemble each other. Both fields $\omega$ and $j$ are significantly amplified, which makes the dynamics resolved only for relatively short times.
• Figure 5: RMHD: (a) Total energy variation, and (b) kinetic $E_{kin}$ and magnetic $E_{magn}$ energy evolution over time. The total energy $E_{magn} + E_{kin}$ is conserved up to a relative error of about $10^{-5}$. The magnetic and kinetic energy components are redistributed with a tendency towards equipartition.