Alfv\'en waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics

TL;DR

This paper numerically studies two-dimensional ideal Euler and MHD flows with finite modes, revealing energy spectra behaviors, absence of dissipative ranges, and effects of magnetic fields on small-scale formation and thermalization.

Contribution

It provides new insights into the dynamics of 2D ideal Euler and MHD flows with high-resolution simulations, highlighting spectral properties and thermalization processes.

Findings

Energy spectra follow a $k^{-3/2}$ scaling at intermediate scales.

No dissipative range observed in 2D Euler or MHD flows.

Magnetic fields slow down small-scale formation and thermalization.

Abstract

We investigate numerically the dynamics of two-dimensional Euler and ideal magnetohydrodynamics (MHD) flows in systems with a finite number of modes, up to $4096^2$, for which several quadratic invariants are preserved by the truncation and the statistical equilibria are known. Initial conditions are the Orszag-Tang vortex with a neutral X-point centered on a stagnation point of the velocity field in the large scales. In MHD, we observe that the total energy spectra at intermediate times and intermediate scales correspond to the interactions of eddies and waves, $E_T(k)\sim k^{-3/2}$. Moreover, no dissipative range is visible neither for Euler nor for MHD in two dimensions; in the former case, this may be linked to the existence of a vanishing turbulent viscosity whereas in MHD, the numerical resolution employed may be insufficient. When imposing a uniform magnetic field to the flow, we observe a lack of saturation of the formation of small scales together with a significant slowing-down of their equilibration, with however a cut-off independent partial thermalization being reached at intermediate scales.