value of $\hat{P}$ uniformly equal to $0.01$. We perform here additional runs with a uniform pressure in the domain, respectively equal to $[4\times 10^{-2}, 10^{-2}, 10^{-3}, 10^{-4}]$. The non-dimensional density is kept constant between the runs, $\hat{\rho}=1$, as is the volume magnetic field, $\hat{B}_v=1$. The plasma $\beta$ is therefore given by $\beta= 2\hat{\mu}\hat{P}/\hat{B}_v=8\pi\hat{P}$. The different runs correspond to the respective plasma $\beta$ values of $[1.0, 0.25, 0.025, 0.0025]$ The non-dimensional Alfvén velocity, $\hat{c}_A=\hat{B}_V/\sqrt{\hat{\mu} \hat{\rho}}=(4\pi)^{-1/2}\simeq0.28$ is therefore constant between the runs, which allows us to compare directly the evolutions of the four cases. The non-dimensional sound speed $\hat{c}_S= (\gamma \hat{P}/\hat{\rho})^{1/2}=(5\hat{P}/3)^{1/2}$ decreases with $\hat{P}$ (and $\beta$), adopting the following values $[0.26,0.13,0.041,0.013]$. The parameters used in the four simulations are summarized in Table \[Tab:ScalNonDim\].

    $\beta$     $\hat{P}$    $\hat{c}_A$   $\hat{c}_S$
  ------------ ------------ ------------- -------------
    $ 1.0 $      $ 0.04 $     $ 0.28 $      $ 0.26 $
    $ 0.25 $     $ 0.01 $     $ 0.28 $      $ 0.13 $
   $ 0.025 $    $10^{-3} $    $ 0.28 $      $ 0.041 $
   $ 0.0025 $   $10^{-4} $    $ 0.28 $      $ 0.013 $

  : Characteristic of different simulations in non-dimensional units: $\beta$, pressure $\hat{P}$, Alfvén speed $\hat{c}_A$, sound speed $\hat{c}_S$. []{data-label="Tab:ScalNonDim"}

These four simulations allow us to simulate 