The World’s Largest Supersonic, Magnetohydrodynamic Turbulence Simulation

**Principal Investigator:**

James R. Beattie and Prof. Christoph Federrath

**Affiliation:**

Research School of Astronomy and Astrophysics, Australian National University

**Local Project ID:**

pn73fi

**HPC Platform used:**

SuperMUC at LRZ

**Date published:**

Supersonic, magnetised turbulence is ubiquitous in the interstellar medium of galaxies. Unlike incompressible turbulence, supersonic turbulence is not scale-free. The scale that marks the transition from supersonic to subsonic turbulence is the so-called sonic scale, which in the context of star formation may define the critical value for which regions inside of molecular gas clouds collapse under their own gravity to form stars. The sonic scale was recently measured for the first time within a hydrodynamical supersonic turbulence simulation using a grid resolution of 10,048^{3} in [1]. This is the only calculation with sufficient resolution to separate the energy injection scale, supersonic cascade, sonic scale, subsonic cascade and dissipation scale of the turbulence. Whilst this calculation brought great insight into the exact position of the sonic scale and the nature of supersonic turbulence, the influence that magnetic fields have on the sonic scale is more or less unknown, yet is of tantamount importance for understanding the nature of supersonic, magnetized turbulence in the interstellar medium of galaxies.

In this LRZ project, we run the first supersonic, magnetised turbulence simulation that is sufficiently resolved to measure the exact position of the (magneto)-sonic scale, and the supersonic and subsonic energy cascades, to determine how magnetic fields affect the turbulence. We do this by running a magnetohydrodynamical (MHD) turbulence simulation with 10,080^{3} grid cells, distributed over almost 140,000 compute cores and run for over 60 million compute-core hours on SuperMUC-NG.

**Numerical and high-performance computing methods**

We use a modified version of the MHD code flash [2]. Our code uses a highly-optimised, hybrid-precision, second-order, positivity-preserving MUSCL-Hancock HLL5R Riemann scheme [3] to solve the ideal, isothermal, compressible magnetohydrodynamic equations in three dimensions with stochastic, large-scale forcing to drive finite time-correlated supersonic turbulence.

We use a block-structured parallelization, with each 3D computational block distributed onto one single compute core. Each block contains 168x210x210 computational cells, which we pass to each of the 138,240 compute cores (2880 compute nodes on SuperMUC-NG) to give a total grid resolution of 10,080^{3} grid cells in each of the three dimensions, and 3.0PB of data products in total. To ensure numerical convergence of the plasma statistics, such as the turbulent energy spectra, we also perform simulations at 1,152^{3} , 2,520^{3} and 5,040^{3} grid resolutions.

In Figure 1 we show the kinetic turbulent energy spectra, *E*_{kin}*(k)*, where *k=2π/l* is the wavenumber, at the four different resolutions as indicated by the line colour. The different regimes of turbulence are shown by the background colour, with the magneto-sonic transition traced in blue.

We see two distinct regimes of turbulence emerging in the high-resolution simulations. On small *k* (large spatial scales, coloured green) the turbulence is dominated by networks of interacting shocks, and has an energy spectrum consistent with Burgers turbulence, *E*_{kin}*(k)**∝**k*^{ -2} . On large *k* (small spatial scales, coloured yellow) the energy cascade resembles the incompressible magnetised turbulence Iroshnikov-Kraichnan spectrum, *E*_{kin}*(k)∝**k *^{-3/2} , corresponding to magnetised vortices, which exchange energy and cascade the energy to smaller and smaller spatial scales.

In Figure 2 we show a two-dimensional slice of the three-dimensional gas density in the simulation, revealing strong, shocked gas (coloured red to orange for the most extreme) and deep rarefactions (coloured blue and green for the most under-dense), typical of supersonic turbulence in the interstellar medium. The local magnetic field is shown with a line integral convolution filter, highlighting the intricate, complex, small-scale structure of the field.

We are currently evolving the 10,080^{3} simulation further on SuperMUC-NG to produce robust measurements of the turbulence on, above, and below the magneto-sonic scale, across multiple correlation times of the turbulent forcing.

[1] C. Federrath et al., Nature Astronomy, 5 (2021) 365-371.

[2] Fryxell et al., Astrophysical Journal Supplement, 131 (2009) 273-334.

[3] Waagan, Federrath & Klingenberg, Journal of Computational Physics, 230 (2011) 3331-3351