Federrath et al. (2010)

The big idea¶

Supersonic turbulence stirs molecular-cloud gas into an enormous range of densities. How wide that range is — the variance of the density PDF — is the single most important input to analytic theories of star formation (the IMF, the SFR, the dense-gas fraction). Federrath+2010 (FK10) show that this width is not set by the Mach number alone: it depends just as strongly on how the turbulence is driven. Purely solenoidal (divergence-free) forcing and purely compressive (curl-free) forcing, at the same Mach number, produce density PDFs whose standard deviations differ by a factor of ~3. This is encoded in a single forcing parameter $b$.

Core relations¶

Work in the logarithmic density $s = \ln(\rho/\langle\rho\rangle)$ (FK10 Eq. 1). For driven isothermal supersonic turbulence the volume-weighted PDF of $s$ is close to a lognormal (Eq. 10):

Mass conservation ($\int e^{s} p_s\,ds = \langle\rho\rangle/\langle\rho\rangle = 1$) fixes the mean in terms of the variance (Eq. 11):

The density-dispersion–Mach relation is stated in two equivalent forms. The linear form for the (non-log) density (Eq. 18, after Padoan, Nordlund & Jones 1997; Passot & Vázquez-Semadeni 1998):

and — assuming the lognormal (Eq. 10) — its logarithmic counterpart (Eq. 19), the relation progenax actually uses:

with the same parameter $b$. Here $\mathcal{M}=\sigma_v/c_s$ is the rms sonic Mach number.

The forcing parameter $b$¶

$b$ measures the fraction of compressive (longitudinal) power in the driving:

FK10 §3.6 establishes $b$ as a smooth function of the forcing parameter $\zeta$ and reconciles the earlier disagreement (Padoan+1997 found $b\approx0.5$; Passot & Vázquez-Semadeni 1998 found $b\approx1$) as different points along this $b(\zeta)$ curve.

Departures from lognormality (intermittency)¶

FK10 emphasise that the PDF is not perfectly lognormal: there are non-Gaussian skewness and kurtosis in the wings, caused by intermittency (rare strong shocks and rarefactions). They model these with a skewed lognormal (Azzalini 1985; Eq. 14) and a 4th-order expansion (Eq. 17). This is the reason a purely Gaussian/lognormal description — and, by extension, a Gaussian random field — captures the variance but not the coherent filaments and sheets of real supersonic turbulence.

Use in progenax¶

• cluster/turbulence.py — sigma_ln_rho_from_mach implements Eq. 19; b_from_environment interpolates $b\in[1/3, \sim0.7]$ with density (a tentative mapping, not from FK10).

• experimental/gravoturb/theory/density_pdf.py — sigma_s_squared(mach, b) is exactly Eq. 19; it is the entry point of the BM19 gravoturbulent density PDF (Burkhart & Mocz (2019)).

Notes¶

• Citation correction. The σ_s²–Mach relation is FK10 Eq. 19 (with Eq. 18 the linear $\sigma_\rho/\langle\rho\rangle=b\mathcal{M}$). The progenax docstring previously attributed it to “Eq. 14”, which is in fact the Azzalini skewed-lognormal PDF — a citation bug fixed in the 2026-06 clean-room pass.

• $b$ is a driving diagnostic, not a free knob. It should sit in $[1/3, 1]$; values outside that range are unphysical for isothermal turbulence.

• Intermittency is real and unmodelled here. The lognormal (and any Gaussian random field built on it) reproduces $\sigma_s^2$ but not the non-Gaussian wings / coherent structures. This is a known limitation of the FDF field realisation.

• The column-density PDF (observable) has a smaller dispersion than the volumetric PDF because line-of-sight integration averages out fluctuations (FK10 §3.5).