Introduction

This page summarizes the results of numerical simulations for isotropic turbulence. The code is public. You can download it from github.

The character of the turbulence depends on the dimension and magnetic fields (see table 8.1 of Biskamp (2008)).

The meaning of the variables are as follows. The subscript $k$ infers the wave number dependence, i.e., those are spectrum.

$E_k$

$\Omega_k$

$H_k^C$

$A_k$

$H_k^K$

$H_k^M$

kinetic energy

enstrophy

cross helicity

magnetic potential

kinematic helicity

magnetic helicity

$E=\rho \vec{v}^2$

$\Omega = (\vec{\nabla} \times \vec{v})_z^2$

$H^C=\vec{v} \cdot \vec{B}$

$A=(-A_z)^2$

$\vec{v} \cdot \vec{\nabla} \times \vec{v}$

$\vec{A} \cdot \vec{B}$

References

Biskamp (2008)

Summary

First let us see the results of 2D simulations. Comparing the spectrum of kinetic energy between hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations, the HD simulation shows an excess in low wavenumber, i.e., the energy is concentrated in lower $k$, large scale. This would be the consequence of the inverse cascading feature. In both simulation, the initial structure is $k\sim10$.

The resolution is 512x512.

This feature could also be seen in the profiles of vorticity. The small scale structure of the vorticity remains in MHD simulation while it disappears in HD simulations.

Vorticity in hydrodynamic simulation. The resolution is 512x512.

Vorticity in magneto-hydrodynamic simulation. The resolution is 512x512.

The kinetic energy and enstrophy in hydrodynamic simulation. The resolution is 512x512.

In hydrodynamic simulation, the kinetic energy spectrum shows inverse cascade and enstrophy spectrum shows direct cascade.

The spectrums in magneto-hydrodynamics. The resolution is 512x512.

In magneto-hydrodynamic simulation, the magnetic potential shows inverse cascading feature while other shows direct cascade.

Next, in 3D simulations, the magnetic energy spectrum shows inverse cascade. In this case, the input perturbation has $k\sim2$. So it would not be the inverse cascade but anyway, the large scale component of magnetic fields remains.

The resolution is 200^3.