Magnetized decaying turbulence in the weakly compressible Taylor-Green vortex

Magnetohydrodynamic (MHD) turbulence affects both terrestrial and astrophysical plasmas. The properties of magnetized turbulence must be better understood to more accurately characterize these systems. This work presents ideal MHD simulations of the compressible Taylor-Green vortex under a range of initial subsonic Mach numbers and magnetic field strengths. We find that regardless of the initial field strength, the magnetic energy becomes dominant over the kinetic energy on all scales after at most several dynamical times. The spectral indices of the kinetic and magnetic energy spectra become shallower than $k^{-5/3}$ over time and generally fluctuate. Using a shell-to-shell energy transfer analysis framework, we find that the magnetic fields facilitate a significant amount of the energy flux and that the kinetic energy cascade is suppressed. Moreover, we observe nonlocal energy transfer from the large-scale kinetic energy to intermediate and small-scale magnetic energy via magnetic tension. We conclude that even in intermittently or singularly driven weakly magnetized systems, the dynamical effects of magnetic fields cannot be neglected.

Figure 1

Slices of sonic Mach number (left) and magnetic pressure (right) at $t=0.77T$ and $5.16T$ in the $xy$ plane through $z=\frac{\pi }{2}L$, with streamlines on the left showing the direction of flow and streamlines on the right showing the direction of the magnetic fields, plotting only the first quadrant from the Ms0.2_Ma10 simulation, demonstrating the transition of the flow into turbulence.

Figure 2

Mean energies over over time in the top row with kinetic energy (solid blue), magnetic energy (solid orange), the sum of kinetic and magnetic energies (solid green), and the change in thermal energy since the simulation start (solid red), and dimensionless numbers over time in the bottom row with RMS sonic Mach number ${\mathcal{M}}_s$ (blue), Alvénic Mach number ${\mathcal{M}}_A$ (orange), and plasma beta $\beta$ (green) for the Ms0.2 simulations. Energy over time from the simulation from Fig. 3 in Pouquet et al. [13] (adjusted to the normalization used here), which matches the setup of the Ms0.2_Ma1 simulation, is shown with dashed lines in the upper left panel for reference. Energies and mach numbers for all nine simulations are shown in the Supplemental Material [21].

Figure 3

Kinetic energy spectra (in solid blue) and magnetic energy spectra (in solid orange) compensated by $k^{4/3}$, with black dashed lines showing the power law fit to the spectral to obtain a spectral index. In the left column we show the Ms0.2_Ma1 simulation, in the middle column we show the Ms0.2_Ma3.2 simulation, and in the right column we show the Ms0.2_Ma10 simulation. In the top row we show all simulations at $t=0.77T$, in the middle row we show the three simulations at different times ($t=1.29, t=1.81T, t=1.81T$) when the simulations are displaying interesting behavior discussed in Secs. 3b2 and 3b1, and in the bottom row we show all simulations at $t=5.16T$ when the initial flow has completely decayed into turbulence and both energy spectra fluctuate around a $k^{-4/3}$ spectrum.

Figure 4

The kinetic energy (top) and magnetic energy (bottom) at wave numbers $k=8,22,64,128$ plotted separately in different colors versus time, where the energy at each wave number has been compensated by $k^{4/3}$ to make them comparable. In the left column we show the Ms0.2_Ma1 simulation, in the middle column we show the Ms0.2_Ma3.2 simulation, and in the right column we show the Ms0.2_Ma10 simulation. Energy at the smallest length scales in both reservoirs saturates at $t\simeq 1T, t\simeq 1.5T$, and $t\simeq 2.5$ in the Ms0.2_Ma1, Ms0.2_Ma3.2, and Ms0.2_Ma10 simulations, respectively, showing approximately when the turbulence has developed at all scales.

Figure 5

Evolution of the spectral indices of the kinetic (blue), magnetic (orange), and sum of kinetic and magnetic energy (green) spectra over time for the Ms0.2 simulations. The slope is computed from a least squares fit of the energy spectra limited to wave numbers $k\in [10,32]$ which is approximately the inertial range. Shaded bands show how the fitted slope differs if a range $k\in [8,34], k\in [10,32]$, or $k\in [12,30]$ is used. Note that the spectral index using the range $k\in [10,32]$ is not guaranteed to be bounded by the spectral indices obtained using $k\in [8,34], k\in [10,32]$ and $k\in [12,30]$, which is especially evident in the Ms0.2_Ma3.2 and Ms0.2_Ma10 simulations from $t\simeq 2T$ to $4T$. Horizontal dashed lines show $-\frac{4}{3}$ and $-\frac{5}{3}$ spectral indices. The slope is only shown after $t=1T$ as the initial flow conditions dominate the spectra at early times, leading to steep spectra. We include the spectral indices versus time for all nine simulations in the Supplemental Material [21].

Figure 6

Shell-to-shell energy transfer plots for the energy transfer within the kinetic (left) and magnetic (right) energy reservoirs via advection and compression at $t=0.77T$ (top) and $t=5.16T$ (bottom) from the simulations with Ms0.2_Ma1, showing the development of the kinetic and magnetic turbulent cascades. Annotations on the figure highlight key features of the energy transfer that are characteristic of a developing turbulence cascade. Each bin shows the flux of energy from shell $Q$ to shell $K$, where orange with white circles showing a positive flux of energy, so that $K$ is gaining energy, and purple with white x's showing a negative flux, so that $K$ is losing energy. The energy flux in each bin is normalized by $\varepsilon ={max}_{Q,K}\left|{\mathcal{T}}_{XY}(Q,K)\right|$ so that a higher $\varepsilon$ means a higher energy flux. The solid black line shows equivalent scale transfers. As the turbulent cascade develops in the magnetic and kinetic energy reservoirs, more energy transfers along the diagonal fill out the energy spectrum down to numerical dissipation scales.

Figure 7

Shell-to-shell energy transfer plots for the energy transfer within the kinetic (top) and magnetic (bottom) energy reservoirs via advection and compression at $t=1.29T$ from the Ms0.2_Ma1 simulation, showing a transient inverse cascade within the magnetic energy reservoir (on all scales $K,Q\lesssim 100$) and kinetic energy reservoir (on large scales $K,Q\lesssim 16$). Annotations show where along the diagonal the inverse cascade is present.

Figure 8

Shell-to-shell energy transfer plots for the energy transfer from kinetic to magnetic energy via magnetic tension at $t=1.81T$ from the Ms0.2_Ma10 simulation, showing the nonlocal energy transfer from large kinetic scales to many smaller magnetic scales. Annotations show where the nonlocal transfer is present.

Figure 9

Integrated energy flux over time from kinetic to magnetic energy via tension from larger wave numbers to smaller nonlocal wave numbers (purple), from larger wave numbers to smaller local wave numbers (blue), between equivalent wave numbers (green), from smaller wave numbers to larger local wave numbers (orange), and from smaller wave numbers to larger nonlocal wave numbers (red) in the Ms0.2 simulations. We normalize the energy flux in each panel so that the absolute maximum of all of the flux bins is 1.0, where $\varepsilon$ is the normalization factor use in each panel. Comparisons of the relative strength of energy fluxes in different simulations must consider $\varepsilon$. The inset plot in the lower right panel shows the color coded regions that are integrated to calculate each line at a single time for the same shell-to-shell transfer from Fig. 8. Solid lines show the integrated flux if “local” wave numbers as defined as five logarithmic bins away from the equivalent wave number. The shaded regions show the integrated flux if four or six bins are used, showing that the behavior is robust if the range “local” wave numbers is defined closer or further away from transfer between equivalent scales. We include the integrated flux from kinetic to magnetic energy via tension for all nine simulations in the Supplemental Material [21].

Figure 10

Integrated energy flux over time within the kinetic energy (top) and within the magnetic energy (bottom) from larger wave numbers to smaller nonlocal wave numbers (purple), from larger wave numbers to smaller local wave numbers (blue), between equivalent wave numbers (green), from smaller wave numbers to larger local wave numbers (orange), and from smaller wave numbers to larger nonlocal wave numbers (red) in the Ms0.2_Ma1 simulation. The inset plot in the lower middle panel demonstrates the color coded regions that are integrated to calculate each line at $t=1.29T$ from the shell-to-shell transfer from Fig. 7. Solid lines show the integrated flux if “local” wave numbers as defined as five logarithmic bins away from the equivalent wave number. The results change very little if four or six bins are used. We include the integrated flux within the kinetic energy and magnetic energy for all nine simulations in the Supplemental Material [21].

Figure 11

Cross-scale flux within the kinetic energy (blue line), within the magnetic energy (orange line), and from kinetic to magnetic energy via tension (green line) in the three Ms0.2 simulations across columns and at dynamical time $t=0.77T$ (top) and later at dynamical time $t=5.16T$. Note that the cross-scale fluxes at later times are an order of magnitude less than early cross-scale fluxes. Positive values of this quantity denote energy transfer from larger to smaller scales.