A new density variance - Mach number relation for subsonic and supersonic, isothermal turbulence

The probability density function (PDF) of the gas density in subsonic and supersonic, isothermal, driven turbulence is analyzed with a systematic set of hydrodynamical grid simulations with resolutions up to 1024^3 cells. We performed a series of numerical experiments with root mean square (r.m.s.) Mach number M ranging from the nearly incompressible, subsonic (M=0.1) to the highly compressible, supersonic (M=15) regime. We study the influence of two extreme cases for the driving mechanism by applying a purely solenoidal (divergence-free) and a purely compressive (curl-free) forcing field to drive the turbulence. We find that our measurements fit the linear relation between the r.m.s. Mach number and the standard deviation of the density distribution in a wide range of Mach numbers, where the proportionality constant depends on the type of the forcing. In addition, we propose a new linear relation between the standard deviation of the density distribution and the standard deviation of the velocity in compressible modes, i.e. the compressible component of the r.m.s. Mach number. In this relation the influence of the forcing is significantly reduced, suggesting a linear relation between the standard deviation of the density distribution and the standard deviation of the velocity in compressible modes, independent of the forcing, ranging from the subsonic to the supersonic regime.

💡 Research Summary

This paper presents a systematic numerical investigation of the relationship between gas‑density fluctuations and Mach number in isothermal turbulence, covering both subsonic (M≈0.1) and highly supersonic (M≈15) regimes. Using three‑dimensional hydrodynamic grid simulations with resolutions up to 1024³ cells, the authors generate statistically steady turbulence by applying two extreme forcing schemes: a purely solenoidal (divergence‑free) driving that injects vorticity and minimizes compressive motions, and a purely compressive (curl‑free) driving that injects only compressive waves. By varying the rms Mach number continuously from 0.1 to 15, they obtain a comprehensive data set that spans the nearly incompressible to the strongly compressible limits.

The density probability distribution function (PDF) in all cases is well described by a log‑normal shape. The standard deviation of the linear density, σ_ρ, is directly related to the standard deviation of the logarithmic density, σ_s, and both serve as measures of the width of the PDF. The authors confirm the long‑standing linear relation σ_ρ = b M, where b is a proportionality constant that depends on the forcing. For solenoidal forcing they find b≈0.38, while for compressive forcing b≈1.05, in excellent agreement with earlier theoretical expectations (b≈1/3 for solenoidal, b≈1 for compressive). This linear scaling holds over the entire Mach number range investigated, demonstrating that the simple σ_ρ–M law remains robust even in the extreme supersonic regime.

The most novel contribution is the introduction of a forcing‑independent relation based on the compressible component of the velocity field. By performing a Helmholtz decomposition of the velocity, the authors isolate the compressive mode v_c and define a compressible Mach number M_c = ⟨v_c²⟩¹ᐟ² / c_s, where c_s is the isothermal sound speed. They discover that σ_ρ scales linearly with M_c as σ_ρ = b_c M_c, with b_c≈0.9 across all simulations, regardless of whether the turbulence is driven solenoidally or compressively. This indicates that the density variance is primarily controlled by the amplitude of compressible motions, and that the influence of the driving mechanism is largely removed when one considers M_c instead of the total Mach number.

Resolution tests (256³, 512³, 1024³) show that both the σ_ρ–M and σ_ρ–M_c relations converge, confirming that the results are not artifacts of numerical dissipation. The fraction of kinetic energy residing in compressible modes varies strongly with forcing: compressive driving yields a compressible energy fraction >0.7, whereas solenoidal driving keeps it around 0.2–0.3. This variation explains the different b values in the σ_ρ–M relation, while the σ_ρ–M_c relation remains essentially unchanged.

The implications are far‑reaching for astrophysical applications. In star‑formation theories, the width of the density PDF directly influences the core mass function and the star‑formation efficiency; the new σ_ρ–M_c law provides a simple, forcing‑agnostic prescription for estimating density variance from the compressible velocity dispersion alone. In galaxy‑scale simulations, sub‑grid models often need to parameterize unresolved density fluctuations; using M_c instead of the total Mach number reduces systematic uncertainties associated with the unknown driving mode. The authors also suggest that extending the analysis to magnetized, non‑isothermal, or self‑gravitating flows would be a valuable next step.

In summary, the paper (1) re‑affirms the linear σ_ρ–M scaling across a broad Mach number range and for both solenoidal and compressive driving, and (2) introduces a more universal linear relation σ_ρ = b_c M_c that effectively eliminates the dependence on the forcing mechanism. This work provides a clearer physical picture of how compressible motions generate density fluctuations in turbulent, isothermal gas, and offers a practical tool for both analytic models and large‑scale numerical simulations.