The statistics and structure of dissipation in subsonic and supersonic turbulence

Abstract

Turbulence plays a critical role in the atmosphere, oceans, engineering, and astrophysics. The dissipation (heating) induced by turbulent flows is particularly important for the thermodynamics and chemistry of interstellar clouds, yet its structure and statistics remain poorly understood. Using high-resolution turbulence simulations with controlled explicit viscosity, we study the kinetic energy dissipation rate, $\varepsilon_{\mathrm{kin}}$, across subsonic and supersonic regimes. We find that dissipation lags large-scale kinetic energy injection events by $1.64\pm0.21$ and $0.48\pm0.07$ turbulent turnover times in subsonic and supersonic turbulence, respectively. Correlations show $\varepsilon_{\mathrm{kin}}\propto\vert\nabla\times\mathbf{v}\vert^2$ (vorticity squared) in the subsonic regime, where density fluctuations are negligible, while in the supersonic regime dissipation is primarily correlated with density, $\varepsilon_{\mathrm{kin}}\proptoρ^{3/2}$. A spectral analysis demonstrates that achieving numerical convergence of $\varepsilon_{\mathrm{kin}}$ across all scales is challenging, especially in the subsonic case, even at $2048^3$ resolution. Nonetheless, subsonic dissipation is clearly localised on small vorticity-dominated scales, while supersonic dissipation spans many scales, combining elongated, thin shocks with small-scale vorticity. Finally, we determine the fractal dimension of $\varepsilon_{\mathrm{kin}}$. In the subsonic regime, intense dissipation is predominantly organised in flattened vortex filaments embedded in thin shearing layers on small scales, becoming more volume-filling at larger scales. In the supersonic regime, $\varepsilon_{\mathrm{kin}}$ exhibits a fractal dimension between 1 and 2 across nearly all scales, likely reflecting shock surfaces and their intersections forming filaments.

Figures (8)

• Figure 1: Slices through the kinetic energy dissipation rate $\varepsilon_{\mathrm{kin}}$ (top panels) and the gas density (bottom panels), in the subsonic regime (${\mathcal{M}}=0.2$; left-hand panels) and the supersonic regime (${\mathcal{M}}=5$; right-hand panels), at $t=5\,t_{\mathrm{turb}}$, i.e., in the fully-developed state of turbulence, for the simulations with $2048^3$ grid cells (cf. Tab. \ref{['tab:sim_params']}). We see that for subsonic turbulence, $\varepsilon_{\mathrm{kin}}$ is more volume filling than in supersonic turbulence, and the dissipation is organised into relatively thick elongated structures. In contrast, the supersonic case shows very thin high-dissipation structures, likely associated with shocks, as reflected in the density field.
• Figure 2: Time evolution of ${\mathcal{M}}$ (top panels) and $\varepsilon_{\mathrm{kin}}$ (bottom panels) for the subsonic regime (left-hand panels) and supersonic regime (right-hand panels) for the $256^3$, $512^3$, $1024^3$, and $2048^3$ simulation resolutions (see legend). Additionally, in the bottom panels, the injection rate $\varepsilon_{\mathrm{inj}}$, is shown as a solid blue line for the $2048^3$ runs, noting that $\varepsilon_{\mathrm{inj}}$ is prescribed by the turbulence driving (cf. Sec. \ref{['sec:turbdriv']}), and therefore identical across simulations with different resolution. The comparison of $\varepsilon_{\mathrm{inj}}$ with $\varepsilon_{\mathrm{kin}}$ together with the resolution dependence of $\varepsilon_{\mathrm{kin}}$ emphasises the need for sufficiently high resolution (here $\gtrsim1024^3$) to converge to the physical dissipation rate set by the viscosity $\nu$ and associated target Reynolds number $\mathrm{Re}=2500$. We also note an apparent time delay between the injection and dissipation rate, which is quantified in Fig. \ref{['fig:time_correlation']}.
• Figure 3: Correlation probability distribution functions between the dissipation rate $\varepsilon_{\mathrm{kin}}$, and density $\rho$ (top panels) or vorticity $\nabla\times\mathbf{v}$ (bottom panels) in subsonic (${\mathcal{M}}=0.2$, left) and supersonic (${\mathcal{M}}=5$, right) turbulence. The dotted lines show approximate power-law trends. In the supersonic regime, $\varepsilon_{\mathrm{kin}}$ correlates with density as $\varepsilon_{\mathrm{kin}}\propto\rho^{3/2}$, consistent with shock-dominated dissipation. In the subsonic regime, dissipation correlates with vorticity as $\varepsilon_{\mathrm{kin}}\propto|\nabla\times\mathbf{v}|^2$, consistent with small-scale vortex dissipation. The same fit is superimposed in the bottom right panel to facilitate direct comparison between the subsonic and supersonic regime, indicating that some dissipation --- in addition to shocks --- stems from vorticity also in the supersonic regime.
• Figure 4: Velocity ($\mathcal{M}$; top panels) and dissipation rate ($\varepsilon_{\mathrm{kin}}$; bottom panels) power spectra for subsonic (left panels) and supersonic (right panels) turbulence. The velocity spectra (top panels) are compensated by $k^{-1.6}$ and $k^{-2.2}$, respectively, to emphasise the approximate self-similar scaling range ('inertial range'). The driving scale at $k=2$ is visible in all spectra, while the viscous dissipation wave number $k_\nu$KrielEtAl2025 is marked as a red vertical band. In the supersonic case the sonic scale $k_\mathrm{s}$FederrathEtAl2021 is indicated as a gold vertical band. The resolution dependence shows that we require $N\gtrsim1024$ for the velocity spectra to converge on scales $k\lesssim k_\nu$ in both the subsonic and supersonic regime. The dissipation spectra (bottom panels) are normalised by $(\propto{\mathcal{M}}^2t_{\mathrm{turb}}^{-1})^2$ to enable direct comparison between regimes. Subsonic dissipation fails to converge even at $2048^3$. However, the peak dissipation --- while not fully converged either --- is relatively stable around $k\sim30$--$50$, indicative of dissipation concentrated on small scales, i.e., relatively large wave numbers close to, but somewhat smaller than $k_\nu$. In contrast, the supersonic spectra converge on scales $k\lesssim k_\nu$ at $\sim1024^3$, with a relatively flat dissipation spectrum. There is some indication of two weak local peaks near $k\sim10$ and $k\sim k_\nu$, which may reflect the combination of elongated shocks extending through large fractions of the domain, and their thin widths. In addition, the supersonic case also has contributions from vortex dissipation on small scales. This demonstrates the fundamentally different dissipation behaviour: subsonic turbulence is dominated by small-scale dissipation, while supersonic turbulence exhibits dissipation across a broader range of scales.
• Figure 5: Fractal dimension analysis of the dissipation rate for subsonic (left) and supersonic (right) turbulence and for different grid resolutions. The shaded regions indicate $1\sigma$ temporal fluctuations. Vertical lines denote the viscous dissipation scale, $\ell_\nu$ (red), and for the supersonic case, the sonic scale, $\ell_\mathrm{s}$ (gold). Power-law fits are shown with slopes annotated in each panel. In the subsonic regime, the measured fractal dimension, $D$, varies strongly with scale. Around $\ell_\nu$, the slope corresponds to $D=1.99\pm0.07$, indicative of sheet-like vortex dissipation structures, while at larger fractions of the box ($\ell/L\in[0.2,0.6]$) we obtain $D=2.56\pm0.18$, consistent with more volume-filling dissipation. In the supersonic regime, the fractal dimension is nearly scale-invariant. At $\ell_\nu$ we measure $D=1.60\pm0.07$, suggesting a mixture of sheet- and filament-like structures, with sheets somewhat dominating. On larger scales the fractal dimension decreases to $D=1.35\pm0.05$, indicating that dissipation becomes increasingly filamentary. This behaviour is consistent with dissipation concentrated at the intersections of strong, long shocks. Resolution dependence is evident in both regimes, but stronger for the subsonic case: convergence requires $N\gtrsim1024^3$ to converge on scales $\ell\gtrsim\ell_\nu$ (cf. Tab \ref{['tab:sim_params']}).