Kim & Ryu (2005)

The big idea¶

The density field of supersonic turbulence is not a passive copy of the velocity field. Kim & Ryu simulate driven isothermal compressible turbulence ($1\lesssim \mathcal{M}_\mathrm{rms}\lesssim10$) and measure the density power spectrum $P_\rho(k)\propto k^{-\beta}$. Their central result: the density spectrum becomes shallower (flatter) as the Mach number rises, because supersonic turbulence sweeps mass into thin sheets, filaments, and ultimately point-like peaks — structures whose Fourier content is dominated by sharp edges (shallow spectra), not by a smooth cascade.

This is the opposite of the intuition you get from the velocity spectrum (Kolmogorov $\to$ Burgers, which steepens from $k^{-5/3}$ toward $k^{-2}$). Conflating the two is a common and consequential mistake.

Core results¶

Let the shell-integrated density spectrum scale as $E_\rho(k)\propto k^{-s}$.

One dimension (compressive only): the profile changes from shock discontinuities at $\mathcal{M}\sim1$ to delta-function peaks at $\mathcal{M}\gg1$:

(These $k^{-2}$ and $k^{0}$ limits are the Burgers shock / strong-shock mass-concentration results of Saichev & Woyczynski 1996.)

Three dimensions (the relevant case), least-squares slopes over the inertial range:

So the 3D density spectrum flattens monotonically with Mach number.

A convention you must get right (E(k) vs P₃D(k))¶

There are two spectra in circulation and they differ by $k^2$:

• the shell-integrated spectrum $E(k)\propto k^{-s}$ (power summed over the $|\mathbf{k}|=k$ shell), and

• the 3D power-spectral density $P_\mathrm{3D}(k)$ (power per Fourier mode), with $E(k) = 4\pi k^2 P_\mathrm{3D}(k)$, hence $P_\mathrm{3D}(k)\propto k^{-(s+2)}$.

Kim & Ryu quote $E_\rho$ (their transonic -1.73 is “close to Kolmogorov $-5/3$”, which is an $E(k)$ statement). The progenax FDF Gaussian random field is built and measured in the $P_\mathrm{3D}$ convention ($P_\mathrm{3D}\propto k^{-\beta}$, see gaussian_field.py). Translating:

i.e. the physical 3D density spectrum sits at $\beta\approx2.5$–3.7 and decreases with Mach — shallower than, and trending opposite to, a pure Kolmogorov/Burgers velocity spectrum.

Use in progenax¶

• This is the correct primary reference for the density power-spectrum slope $\beta$ fed to the FDF GRF (gaussian_field.py) — not the velocity-spectrum theories (Kolmogorov/Burgers).

• Cross-check for Lomax, Bates & Whitworth (2018): Lomax’s FBM field is generated with $\beta=E+2H=3+2H$ (the generating Gaussian field), set by the desired fractal dimension $D=3-H$; the resulting density field then carries the shallower Kim & Ryu spectrum after the nonlinear (lognormal / copula) remap.

Notes¶

• Density ≠ velocity spectrum. An earlier spectral_slope_from_mach (turbulence.py) interpolated between Kolmogorov ($\beta=11/3$) and Burgers ($\beta=4$) — those are velocity slopes, and the implied increase of $\beta$ with Mach is backwards for the density spectrum. Corrected in the 2026-06 grounding pass: spectral_slope_from_mach now returns the Kim & Ryu density slope $\beta(\mathcal{M}) = 3.788 - 1.203\log_{10}\mathcal{M}$ (the least-squares fit of $s+2$ vs $\log_{10}\mathcal{M}$ over the four points above), clipped to $[2,\,11/3]$ (BETA_DENSITY_FLOOR, BETA_KOLMOGOROV in cluster/constants.py).

• The simulations neglect magnetic fields and self-gravity; both modify the slope (Padoan+ 2004 found magnetic fields shallow it further; self-gravity adds power at small scales, the high-density tail of Burkhart & Mocz (2019)).

• The shallow CNM H I power spectrum (Deshpande+2000) vs. the Kolmogorov electron-density spectrum (Armstrong+1995) is explained by this Mach dependence: H I is supersonic ($\mathcal{M}\sim$ few), the WIM transonic.