Turbulence in the Interstellar Medium

Left to itself, interstellar gas would settle into the quiet thermal-pressure balance of the two- and three-phase medium. It does not. Every line profile is far too broad for its temperature: a molecular cloud at $T\sim10\,\mathrm{K}$ has a thermal sound speed of only $c_s\approx0.19\,\mathrm{km\,s^{-1}}$ , yet its CO and dust maps display velocity dispersions of several $\mathrm{km\,s^{-1}}$ . Those non-thermal motions are the fingerprint of turbulence — a chaotic, multi-scale velocity field filling the medium from tens of parsecs to sub-AU scales. Because the flows are compressible and magnetized, this is not the smooth, incompressible turbulence of a laboratory water tank: here shocks, not gentle eddies, do the work. Three ideas organize everything that follows. First, the empirical backbone is the set of Larson relations — the velocity dispersion of a region grows as a power of its size, and cloud masses scale so that clouds sit near virial balance. Second, that velocity–size scaling fixes the turbulent energy spectrum: steeper than Kolmogorov's, close to the Burgers spectrum of a field of shocks. Third, turbulence is a Janus phenomenon — its ram pressure holds clouds up on average, while the very shocks that drain its energy sweep gas into dense, low-Jeans-mass pockets that collapse. Magnetic fields, with an Alfvén speed comparable to the turbulent speed, make the cascade anisotropic and shape the near-lognormal density distribution behind theories of the star-formation rate and the IMF.

Key takeaways

The interstellar medium is turbulent on every observable scale; in cold molecular gas the flows are supersonic ( $\mathcal{M}_s\sim5$ – $20$ ) and trans-Alfvénic ( $\mathcal{M}_A\sim1$ ), so linewidths vastly exceed thermal broadening.
Larson's laws encode this as a velocity–size scaling $\sigma_v\propto\ell^{\,q}$ with $q\approx0.4$ – $0.5$ , matching the Burgers spectrum $E(k)\propto k^{-2}$ rather than the Kolmogorov $k^{-5/3}$ ; the bridge is $\beta=2q+1$ .
Turbulence supports clouds against gravity on large scales while its dissipation in shocks manufactures the dense filaments and cores that become self-gravitating — global support, local collapse.
Supersonic turbulence decays in about one crossing time $t_{\rm cross}=L/\sigma_v$ , so it must be continuously driven — by supernovae, H II regions, galactic shear, accretion, and outflows.
Magnetic fields set the anisotropy (Goldreich–Sridhar, $k_\parallel\propto k_\perp^{2/3}$ ) and, with the Mach number, fix the width of the lognormal density PDF behind star-formation-rate and IMF theory.
Key idea: ISM turbulence is supersonic and magnetized — its kinetic energy cascades from large driving scales to shock-dissipation scales in about one crossing time, supporting clouds against gravity on average while producing the dense fluctuations that locally collapse into stars.

The core insight

ISM turbulence is supersonic and magnetized: its kinetic energy cascades from large driving scales down to shock-dissipation scales in about one crossing time, simultaneously supporting clouds against gravity on average while producing the dense fluctuations that locally collapse into stars.

Where this lives today

Interstellar turbulence is now measured, not just modelled. Wide-field spectral-line surveys — CO with ALMA, dense-gas tracers, H I in nearby galaxies — map velocity dispersions across clouds and recover Larson-type scalings and the velocity power spectrum directly. Herschel dust imaging revealed that molecular clouds are organized into filaments of common width near $0.1\,\mathrm{pc}$ , and their column-density PDFs show the predicted structure: a lognormal body from turbulence plus a power-law tail where self-gravity has taken over — a live diagnostic separating support from collapse. Magnetic fields are read from the same flows via the Chandrasekhar–Fermi method, which combines the scatter of dust-polarization angles (mapped by Planck and SOFIA/HAWC+) with the velocity dispersion to estimate $B$ . The open frontier is the driving — what fraction is powered by supernovae versus gravitational collapse versus feedback, and how the compressive-versus-solenoidal forcing mix (the parameter $b$ ) sets the star-formation rate — now being closed by MHD simulations that resolve the sonic scale.

The physical picture

Picture the interstellar medium as a fluid never allowed to rest. Supernova blast waves and expanding H II regions stir it at large scales — tens of parsecs — and that stirring is chopped into ever-smaller swirls and shocks until the motion is finally thermalized. Between the injection and dissipation scales lies an inertial range across which energy merely passes through, neither made nor destroyed. This conserved flux is the defining signature of a turbulent cascade, and why a single number — the energy injection rate per unit mass — controls the statistics across an enormous range of scales.

Figure

The turbulent cascade in the cold ISM: supernovae and H II regions inject energy at L ≈ 50 pc, a conserved flux ε carries it down an inertial range of supersonic eddies and shocks, and it dissipates in shocks and ambipolar diffusion below ℓ ≈ 0.1 pc — all within one crossing time t_cross = L/σ_v ≈ 50 pc / 5 km s⁻¹ ≈ 10 Myr, with the flow supersonic at every step, ℳ_s ≈ 10

Because the flow is supersonic, neighbouring parcels at different velocities cannot slide smoothly past one another — they collide and shock, sweeping mass into thin sheets and filaments. The density field is therefore not smooth but intermittent: rare dense knots separated by vast rarefied voids. For isothermal supersonic turbulence this has a clean statistical form — a lognormal density distribution whose width is set almost entirely by the sonic Mach number — and that distribution turns a chaotic velocity field into a predictive theory of how much gas sits above the collapse threshold. Turbulence supports a cloud only on average and only above the driving scale. Below it, it does the opposite: it compresses gas locally past the Jeans threshold. This dual character — bulk support, local promotion of collapse — is why one phenomenon both throttles and enables star formation. Threading all of it is a magnetic field whose Alfvén speed is comparable to the turbulent speed; it resists compression across its lines and channels motion along them, so the small-scale eddies are elongated ribbons, not spheres. The second figure shows the empirical law tying a region's size to its velocity dispersion.

Figure

Larson's velocity–size relation σ_v = σ₀ (ℓ / 1 pc)^q with σ₀ ≈ 1 km s⁻¹ and q ≈ ½, plotted log–log across three decades of size: a dense core at 0.1 pc and 0.3 km s⁻¹, the normalization σ₀ = 1 km s⁻¹ at 1 pc, and a clump at 10 pc and 3.2 km s⁻¹ all lie on one line whose slope q ≈ ½ implies the Burgers spectrum E(k) ∝ k⁻² through β = 2q + 1 = 2

The core equations

Larson's velocity–size relation. The empirical anchor: measure a region's one-dimensional velocity dispersion and its size, and the two follow a power law across three decades of scale. The exponent is the diagnostic that points to the cascade's spectrum.

\sigma_v(\ell)=\sigma_0\left(\frac{\ell}{1\,\mathrm{pc}}\right)^{q},\qquad q\approx0.38\text{–}0.5 \qquad (1)

where: $\sigma_v$ — one-dimensional (line-of-sight) velocity dispersion of a region; $\ell$ — characteristic linear size; $\sigma_0\approx1\,\mathrm{km\,s^{-1}}$ — normalization at $1\,\mathrm{pc}$ ; $q$ — scaling exponent, $\tfrac{1}{2}$ for the virialized/Burgers case, $\tfrac{1}{3}$ for Kolmogorov. Bigger regions move faster; the rate encodes the cascade.

The kinetic energy spectrum. In wavenumber space the same information is a power spectrum whose slope $\beta$ is tied to Larson's $q$ by a bridge derived below; the two textbook values are the incompressible Kolmogorov result and the shock-dominated Burgers result.

E(k)\,\mathrm{d}k\propto k^{-\beta}\,\mathrm{d}k,\qquad \beta=2q+1=\begin{cases}5/3 & \text{Kolmogorov}\\ 2 & \text{Burgers}\end{cases} \qquad (2)

where: $E(k)$ — kinetic energy per unit wavenumber; $k\sim1/\ell$ — wavenumber; $\beta$ — spectral slope. Reading it: a steeper spectrum ( $\beta=2$ ) means sharp discontinuities — shocks — at small scales.

The sonic Mach number. Whether the flow is compressible is decided by comparing its speed to the sound speed. In cold molecular gas this ratio is large, which is why the dynamics are shock-dominated.

\mathcal{M}_s=\frac{\sigma_v}{c_s},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}} \qquad (3)

where: $\mathcal{M}_s$ — sonic Mach number; $c_s$ — isothermal sound speed; $T$ — gas temperature ( $\sim10\,\mathrm{K}$ in molecular gas); $\mu\approx2.3$ — mean molecular weight; $m_H$ — hydrogen mass. Reading it: $\mathcal{M}_s\gg1$ ( $5$ – $20$ ) is why ISM turbulence is not Kolmogorov's.

The Alfvén speed and Alfvén Mach number. Magnetic tension sets a second wave speed, and the turbulence is trans-Alfvénic — the two are comparable, so the field is dynamically important, not a passive tracer.

v_A=\frac{B}{\sqrt{4\pi\rho}}\approx1.8\left(\frac{B}{10\,\mu\mathrm{G}}\right)\!\left(\frac{n}{10^2\,\mathrm{cm^{-3}}}\right)^{-1/2}\mathrm{km\,s^{-1}},\qquad \mathcal{M}_A=\frac{\sigma_v}{v_A} \qquad (4)

where: $v_A$ — Alfvén speed; $B$ — field strength; $\rho=\mu m_H n$ — mass density; $n$ — number density; $\mathcal{M}_A$ — Alfvén Mach number ( $\sim1$ means magnetic and turbulent energy densities are comparable, forcing an anisotropic cascade).

The lognormal density PDF. Repeated shocks multiply the local density by random factors, and the product of many independent factors is lognormal. The width of that distribution is fixed by the Mach number and the way the gas is driven.

\sigma_{\ln s}^2=\ln\!\left(1+b^2\mathcal{M}_s^2\frac{\beta_p}{\beta_p+1}\right),\qquad s=\ln(\rho/\bar\rho) \qquad (5)

where: $s$ — logarithmic density contrast; $\sigma_{\ln s}$ — width of the lognormal PDF; $b$ — forcing parameter, $\tfrac{1}{3}$ (solenoidal) to $1$ (compressive); $\beta_p=2c_s^2/v_A^2$ — plasma beta ( $\beta_p\to\infty$ recovers the unmagnetized form). Reading it: stronger, more compressive turbulence widens the PDF and makes more dense gas.

Virial balance with turbulence. On the cloud scale, turbulent kinetic energy replaces thermal energy as the term that opposes gravity in the virial theorem.

E_{\rm turb}\simeq\tfrac{3}{2}M\sigma_v^2,\qquad 2E_{\rm turb}+2E_{\rm therm}+\mathcal{E}_{\rm mag}+W=0 \qquad (6)

where: $E_{\rm turb}$ — turbulent kinetic energy of a cloud of mass $M$ ; $E_{\rm therm}$ — thermal energy (usually negligible in GMCs); $\mathcal{E}_{\rm mag}$ — magnetic energy; $W\approx-\tfrac{3}{5}GM^2/R$ — gravitational energy of a uniform sphere. Near virial balance ( $2E_{\rm turb}\approx|W|$ ) a cloud neither collapses nor disperses.

The crossing time and dissipation rate. Supersonic turbulence is not free — it drains itself over a single large-eddy turnover, setting the pace of everything the cloud does.

t_{\rm cross}=\frac{L}{\sigma_v},\qquad \dot E_{\rm diss}\sim-\frac{E_{\rm turb}}{t_{\rm cross}}\sim-\frac{M\sigma_v^3}{L} \qquad (7)

where: $t_{\rm cross}$ — turbulent crossing (dissipation) time; $L$ — driving scale / cloud size; $\dot E_{\rm diss}$ — kinetic-energy dissipation rate. Reading it: an order-unity fraction of the energy is lost each $t_{\rm cross}$ , so persistent turbulence demands driving.

Goldreich–Sridhar anisotropy. When the turbulence is magnetized, eddies stretch along field lines: motion perpendicular to $\mathbf{B}$ cascades faster than parallel motion, producing scale-dependent anisotropy.

k_\parallel\propto k_\perp^{2/3} \qquad (8)

where: $k_\parallel$ — wavenumber along the local field; $k_\perp$ — wavenumber across it. Small eddies are increasingly elongated along $\mathbf{B}$ .

Derivation ALarson's laws from virial equilibrium

The goal is to show that Larson's two empirical scalings — the velocity–size law $\sigma_v\propto\ell^{1/2}$ and the mass–size law $M\propto\ell^2$ — are not independent accidents but automatic consequences of one assumption: that molecular clouds sit near virial balance at a roughly common surface density. We start from the virial theorem with gravity opposed by turbulent kinetic energy alone, neglect the thermal, magnetic, and surface terms, introduce the observed near-constancy of column density, and read the exponents off. The constant-surface-density assumption is the physical crux, and the derivation shows exactly where it enters.

Step 1. Set twice the turbulent kinetic energy equal to the magnitude of the gravitational energy of a uniform sphere of radius $R\sim\ell$ : $2E_{\rm turb}\approx|W|$ , i.e. $3M\sigma_v^2\approx\tfrac{3}{5}GM^2/R$ . Why: this is the virial statement that the cloud is marginally bound — the single dynamical condition from which everything else flows.

Step 2. Cancel and solve for the dispersion, giving $\sigma_v^2\approx GM/(5R)$ . Why: the velocity dispersion is fixed by the depth of the gravitational potential, the mass-to-radius ratio — it is a statement about gravity, not about the driving of turbulence.

Step 3. Introduce the surface density $\Sigma=M/(\pi R^2)$ , so $M=\Sigma\pi R^2$ and $\sigma_v^2\approx(\pi G/5)\,\Sigma\,R$ . Why: observations find $\Sigma$ roughly constant across clouds (the Larson "third law"), which is what collapses two free variables into one and makes a clean power law possible.

Step 4. With $\Sigma\approx\text{const}$ , read off $\sigma_v\propto R^{1/2}$ (so $q=\tfrac{1}{2}$ ) and $M=\Sigma\pi R^2\propto R^2$ . Why: these are exactly Larson's velocity–size and mass–size relations, recovered to within the observed scatter — and the derivation shows the exponents are dictated by virial gravity plus constant column, not by the turbulence spectrum itself.

Go deeper · Derivation B · 1 minThe bridge from velocity–size scaling to the energy spectrum

Derivation BThe bridge from velocity–size scaling to the energy spectrum

The goal is to derive the relation $\beta=2q+1$ connecting the observable Larson exponent $q$ to the theoretical spectral slope $\beta$ , and thereby to justify that a supersonic, shock-dominated ISM ( $q\approx\tfrac{1}{2}$ ) has a Burgers spectrum $E(k)\propto k^{-2}$ rather than the Kolmogorov $k^{-5/3}$ of incompressible flow. The argument is purely kinematic: it relates the velocity variance measured at scale $\ell$ to an integral of the power spectrum over all smaller scales, then matches the resulting $\ell$ -dependence to Larson's law. No shock dynamics are needed — only the definitions of the spectrum and the size-dependent dispersion.

Step 1. Write the velocity variance at scale $\ell$ as the integrated power from that scale down: $\sigma_v^2(\ell)\sim\int_{1/\ell}^{\infty}E(k)\,\mathrm{d}k$ . Why: the dispersion a telescope measures over a region of size $\ell$ collects the contributions of all fluctuations of that size and smaller — it is a running integral of the spectrum.

Step 2. For a power law $E(k)\propto k^{-\beta}$ with $\beta>1$ , the integral is dominated by its lower limit and gives $\int_{1/\ell}^{\infty}k^{-\beta}\,\mathrm{d}k\propto\ell^{\,\beta-1}$ . Why: when $\beta>1$ the spectrum is red — most of the energy sits at the largest scales in the range — so the integral scales with the largest scale included, namely $\ell$ .

Step 3. Match to Larson: $\sigma_v^2\propto\ell^{\,\beta-1}$ must equal $\ell^{\,2q}$ , hence $\beta=2q+1$ . Why: this is the whole bridge — it converts a measured real-space exponent into a spectral one, so that $q=\tfrac{1}{3}\to\beta=\tfrac{5}{3}$ (Kolmogorov) and $q=\tfrac{1}{2}\to\beta=2$ (Burgers).

Step 4. Insert the observed $q\approx0.4$ – $0.5$ : the ISM spectrum is $\beta\approx1.8$ – $2$ , i.e. Burgers-like. Why: the steepening from $5/3$ to $2$ is the spectral signature of discontinuities — a field of shocks — confirming that supersonic ISM turbulence is qualitatively different from the smooth incompressible cascade.

Go deeper · Derivation C · 1 minThe turbulent Jeans mass in a shock-compressed region

Derivation CThe turbulent Jeans mass in a shock-compressed region

The goal is to quantify how supersonic compression lowers the mass that can gravitationally collapse, making precise the claim that turbulence, while supporting a cloud globally, seeds star formation locally. We take an isothermal shock of Mach number $\mathcal{M}_s$ , use the isothermal Rankine–Hugoniot jump to fix the post-shock density, and re-evaluate the Jeans mass there. The result is a compact factor by which the local collapse threshold drops behind the shock — the engine that turns a supporting flow into a fragmenting one.

Step 1. Recall the isothermal Jeans-mass scaling $M_J\propto c_s^3\rho^{-1/2}\propto\rho^{-1/2}$ at fixed temperature. Why: the Jeans mass is the smallest mass that self-gravity can overwhelm against pressure; at fixed $c_s$ it falls as density rises, because denser gas is easier to bind.

Step 2. Apply the strong isothermal shock jump $\rho_1/\rho_0=\mathcal{M}_s^2$ , from mass and momentum conservation across the front. Why: an isothermal shock cannot store energy as heat, so all of the compression goes into density — the jump is the square of the Mach number, far stronger than the factor of $4$ of an adiabatic monatomic shock.

Step 3. Combine to get $M_{J,\rm post}/M_{J,0}=(\rho_0/\rho_1)^{1/2}=\mathcal{M}_s^{-1}$ . Why: a Mach-10 shock raises the density a hundredfold and drops the local Jeans mass tenfold, so a modest packet of swept-up gas can suddenly exceed its Jeans mass and collapse.

Step 4. Read the consequence: the same shocks that dissipate turbulent energy build the dense, low- $M_J$ pockets where gravity wins locally, even as bulk turbulent pressure holds the cloud up. Why: this is the mechanistic core of turbulent fragmentation — support and collapse are two faces of one supersonic flow, which is why clouds form stars slowly but inevitably.

Limiting cases

The character of the turbulence is set by dimensionless ratios; each limit recovers a familiar regime.

Case	Result	Meaning
Subsonic ( $\mathcal{M}_s\ll1$ )	$E(k)\propto k^{-5/3}$	Nearly incompressible Kolmogorov cascade — relevant to the warm ionized and neutral medium and to diffuse gas probed by electron-density scintillation.
Supersonic ( $\mathcal{M}_s\gg1$ )	$E(k)\propto k^{-2}$	Shock-dominated Burgers turbulence — cold molecular clouds with $\mathcal{M}_s\sim5$ – $20$ ; the steeper spectrum reflects discontinuities, not smooth eddies.
Trans-Alfvénic ( $\mathcal{M}_A\sim1$ )	$v_A=B/\sqrt{4\pi\rho}\sim\sigma_v$	Turbulent and magnetic energy densities comparable; the Goldreich–Sridhar anisotropy $k_\parallel\propto k_\perp^{2/3}$ governs the cascade along versus across $\mathbf{B}$ .
Decaying versus driven	$t_{\rm diss}\sim t_{\rm cross}=L/\sigma_v$	Supersonic turbulence decays in about a crossing time regardless of magnetization; persistent turbulence therefore implies active, ongoing energy sources.

Common trapReading a broad linewidth as a high temperature, and a supersonic flow as a Kolmogorov cascade

Two related traps. First, a broad, non-thermal linewidth is not a temperature: in a $10\,\mathrm{K}$ cloud a dispersion of $2\,\mathrm{km\,s^{-1}}$ is turbulent, and reading it as thermal would imply $T\sim10^3\,\mathrm{K}$ , far too hot for CO-emitting gas. The linewidth measures the dynamical state, not the thermodynamic one. Second, do not import incompressible Kolmogorov intuition wholesale: supersonic ISM turbulence dissipates chiefly in shocks — a Burgers field with $E(k)\propto k^{-2}$ and sharp density discontinuities — not through a smooth $k^{-5/3}$ eddy cascade. Using $k^{-5/3}$ , or treating the density field as mildly fluctuating rather than intermittently lognormal, gets both the spectrum and the dense-gas fraction wrong.

Worked examples

1 · The dynamical state of a giant molecular cloud (fully worked)

A clump of radius $R=10\,\mathrm{pc}$ sits at $T=10\,\mathrm{K}$ . Use Larson's law to estimate its velocity dispersion, classify the flow, and infer its virial mass.

Step 1. Larson's law with $\sigma_0=1\,\mathrm{km\,s^{-1}}$ and $q=\tfrac{1}{2}$ gives $\sigma_v=1\times(10)^{1/2}=\mathbf{3.16\,km\,s^{-1}}$ — already far above any thermal speed.

Step 2. The isothermal sound speed is $c_s=\sqrt{k_BT/\mu m_H}=\mathbf{0.19\,km\,s^{-1}}$ for $T=10\,\mathrm{K}$ , $\mu=2.3$ , so $\mathcal{M}_s=\sigma_v/c_s=3.16/0.19\approx\mathbf{17}$ — highly supersonic, shock-dominated, with support overwhelmingly turbulent, not thermal.

Step 3. From the virial relation $\sigma_v^2\approx GM/(5R)$ , $M\approx5\sigma_v^2R/G$ . With $\sigma_v=3.16\times10^5\,\mathrm{cm\,s^{-1}}$ , $R=10\,\mathrm{pc}=3.09\times10^{19}\,\mathrm{cm}$ , $M\approx5(9.99\times10^{10})(3.09\times10^{19})/(6.67\times10^{-8})=2.3\times10^{38}\,\mathrm{g}$ , i.e. $\mathbf{M\approx1.2\times10^5\,M_\odot}$ — a giant molecular cloud held near virial balance by turbulence, not thermal pressure. σ_v = 3.16 km s⁻¹, ℳ_s ≈ 17, M_vir ≈ 1.2×10⁵ M⊙.

2 · How a shock lowers the local Jeans mass (fully worked)

A Mach-10 isothermal shock passes through the same $10\,\mathrm{K}$ gas. By how much does it compress the gas and drop the local Jeans mass, and why does that matter?

Step 1. The isothermal jump gives a density compression $\rho_1/\rho_0=\mathcal{M}_s^2=10^2=\mathbf{100}$ — a hundredfold enhancement swept up behind a single strong shock, which is why post-shock filaments are so much denser than their surroundings.

Step 2. Since $M_J\propto\rho^{-1/2}$ at fixed temperature, the post-shock Jeans mass is $M_{J,\rm post}=M_{J,0}(\rho_0/\rho_1)^{1/2}=M_{J,0}\,\mathcal{M}_s^{-1}=\mathbf{M_{J,0}/10}$ . The local collapse threshold falls by a factor of ten.

Step 3. Gas that was safely below its Jeans mass in the ambient cloud can now exceed it, so the shock triggers collapse in exactly the regions where turbulence is dissipating. A Mach-10 shock gives ρ₁/ρ₀ = 100 and M_J → M_J/10 — the flow supports globally while fragmenting locally.

3 · The width of the density PDF

Estimate the lognormal PDF width for $\mathcal{M}_s=10$ turbulence with mixed forcing $b=0.4$ ( $\beta_p\to\infty$ ).

Step 1. $\sigma_{\ln s}^2=\ln(1+b^2\mathcal{M}_s^2)=\ln(1+0.16\times100)=\ln(17)=\mathbf{2.83}$ , so $\sigma_{\ln s}\approx\mathbf{1.68}$ — the distribution spans many $e$ -foldings in density. With mean $\bar s=-\sigma_{\ln s}^2/2\approx-1.42$ , gas at contrast $\rho/\bar\rho=100$ sits $\approx3.6\sigma$ above the mean — a rare tail. Raising ℳ_s or b fattens this tail sharply, which is how turbulence sets the star-formation rate: by fixing how much gas lives above the collapse threshold.

Physics of the portrait

Supersonic turbulence is where fluid dynamics, magnetism, and gravity meet in the coldest gas of the Galaxy — a story of borrowed intuition corrected: a laboratory theory of smooth eddies, rebuilt around shocks and magnetic fields once it was clear the ISM obeyed neither.

Origins · 1 minsupersonic turbulence and how this was discovered

supersonic turbulence

the shock-dominated cascade E(k) ∝ k⁻² of the cold interstellar medium · astrophysics of star formation

The theory of turbulence entered astronomy through its incompressible parent. Andrei Kolmogorov's 1941 analysis — energy injected at large scales, cascading with constant flux through an inertial range, dissipating by viscosity — gave the famous $E(k)\propto k^{-5/3}$ spectrum and the dimensional reasoning still underlying every cascade. Around 1951 von Weizsäcker and Chandrasekhar were among the first to argue the interstellar medium is turbulent and to link that to star formation. But the ISM refused to be Kolmogorov's fluid: its flows are supersonic, dissipating in shocks rather than smooth eddies, giving the steeper Burgers spectrum $E(k)\propto k^{-2}$ . The decisive empirical step came in 1981, when Richard B. Larson compiled the velocity–size and mass–size scaling relations, showing the medium is self-similar over decades of scale and hovers near virial balance. In 1994 Vázquez-Semadeni showed supersonic isothermal turbulence yields a lognormal density PDF, and in 1995 Goldreich and Sridhar built the theory of anisotropic magnetized turbulence. By the late 1990s the simulations of Mac Low, Stone, and collaborators showed this turbulence decays in about a crossing time — completing the modern picture of driven, supersonic, magnetized turbulence that both supports clouds and seeds the stars inside them.

History

1941

— Kolmogorov formulates the incompressible turbulence theory and its −5/3 energy spectrum

1981

— Larson finds the velocity–size and mass–size scaling relations for molecular clouds

1994

— Vázquez-Semadeni shows supersonic isothermal turbulence yields a lognormal density PDF

1995

— Goldreich and Sridhar derive the anisotropic MHD turbulent cascade

1998

— Mac Low, Stone and collaborators show supersonic turbulence decays in a crossing time

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 12 (molecular clouds and the Larson relations) · Binney and Tremaine, Galactic Dynamics, 2nd ed. (fluid dynamics and virial arguments for the ISM) · Mo, van den Bosch and White, Galaxy Formation and Evolution, ch. 8–9 (gas physics and star formation) · Dodelson, Modern Cosmology (power spectra and log-normal fields, used here for the density PDF).

Reviews and primary sources. Larson (1981), MNRAS 194, 809 — velocity–size and mass–size relations · Elmegreen and Scalo (2004), ARA&A 42, 211 — interstellar turbulence review · Mac Low and Klessen (2004), Rev. Mod. Phys. 76, 125 — star formation by supersonic turbulence. Key relations used throughout: $\beta=2q+1$ ; virial scalings $\sigma_v\propto\Sigma^{1/2}\ell^{1/2}$ and $M\propto\Sigma\ell^2$ ; isothermal-shock compression $\rho_1/\rho_0=\mathcal{M}_s^2$ (Rankine–Hugoniot). Sound speed $c_s=0.19\,\mathrm{km\,s^{-1}}$ at $10\,\mathrm{K}$ , $\mu=2.3$ . Worked anchor numbers: $\mathcal{M}_s\approx17$ and $M_{\rm vir}\approx1.2\times10^5\,M_\odot$ for the $10\,\mathrm{pc}$ clump; $\sigma_{\ln s}\approx1.68$ for $\mathcal{M}_s=10$ , $b=0.4$ ; $B\approx22\,\mu\mathrm{G}$ for the trans-Alfvénic condition. Normalizations: $\sigma_0\approx1\,\mathrm{km\,s^{-1}}$ at $1\,\mathrm{pc}$ ; $q\approx0.38$ – $0.5$ ; $\mathcal{M}_s\sim5$ – $20$ .

Prerequisites. Molecular Clouds and Giant Molecular Clouds · Virial Theorem for Molecular Clouds · The Jeans Instability and Jeans Mass · Interstellar Polarization and Magnetic Fields · Phases of the Interstellar Medium. Leads to. Thermal Instability and the Two-Phase Medium · CO as a Molecular Gas Tracer and the X-Factor · Interstellar Extinction and Reddening.

Ask the tutor: ask me why a supersonic, magnetized cloud can hold itself up against gravity and still form stars — or to derive Larson's law from virial balance step by step.

Why is interstellar turbulence "supersonic" when laboratory turbulence usually is not?

Because the interstellar medium is both very cold and very lightly pressurized. In a $10\,\mathrm{K}$ molecular cloud the sound speed is only about $0.19\,\mathrm{km\,s^{-1}}$ , yet the observed velocity dispersions are several $\mathrm{km\,s^{-1}}$ , so the Mach number is $5$ – $20$ . The consequence is qualitative: supersonic flows cannot adjust smoothly, so they shock. That is why ISM turbulence has a steeper Burgers spectrum and a strongly intermittent, lognormal density field rather than the gentle Kolmogorov statistics of an incompressible fluid.

If turbulence supports clouds against gravity, how does it also cause stars to form?

It does both, at different scales. Above the driving scale, turbulent ram pressure $\rho\sigma_v^2$ acts like a stiff effective pressure that opposes self-gravity and keeps most of the mass from collapsing at once — which is why clouds convert only a few percent of their gas into stars per free-fall time. Below the driving scale, the same supersonic flow does the opposite: colliding streams shock and sweep gas into dense cores whose density is raised by up to $\mathcal{M}_s^2$ , dropping the local Jeans mass and tipping those pockets into collapse. Support is a large-scale, statistical effect; triggering is a small-scale, local one — and they coexist in one flow.

What exactly are the Larson relations, and why do they matter?

They are three empirical scaling laws found by Richard Larson in 1981: the velocity dispersion grows with region size as $\sigma_v\propto\ell^{\,q}$ with $q\approx0.5$ ; the mass grows as $M\propto\ell^2$ ; and clouds hover near virial balance at a roughly common surface density. They matter because they compress the hierarchical ISM into a few power laws, diagnose the cascade (the exponent $q$ fixes the spectral slope through $\beta=2q+1$ ), and let observers estimate a cloud's mass from its size and linewidth. As shown above, all three follow from virial balance plus constant column density.

Why does supersonic turbulence produce a lognormal density distribution?

Because density is built up multiplicatively. As gas moves through a turbulent field it is compressed and rarefied by many successive, roughly independent shocks, each multiplying its density by some factor. The logarithm of a product of many independent positive factors tends, by the central limit theorem, to a normal distribution — so $\ln(\rho/\bar\rho)$ is Gaussian and $\rho$ is lognormal. The width of that Gaussian, $\sigma_{\ln s}\approx\sqrt{\ln(1+b^2\mathcal{M}_s^2)}$ , grows with the Mach number and with how compressive the forcing is. This lognormal is the foundation of analytic theories for the star-formation rate and the IMF, because it fixes how much gas sits in the dense, collapsing tail.

How do magnetic fields change the picture, and how strong are they?

Interstellar fields are dynamically important: their Alfvén speed $v_A=B/\sqrt{4\pi\rho}$ is comparable to the turbulent speed, so the turbulence is trans-Alfvénic ( $\mathcal{M}_A\sim1$ ). A field that strong resists compression across its lines and channels motion along them, making the cascade anisotropic — eddies stretch along $\mathbf{B}$ as $k_\parallel\propto k_\perp^{2/3}$ — and somewhat narrowing the density PDF. Typical molecular-cloud fields are tens of microgauss: a trans-Alfvénic estimate at $n=10^3\,\mathrm{cm^{-3}}$ , $\sigma_v=1\,\mathrm{km\,s^{-1}}$ gives $B\approx22\,\mu\mathrm{G}$ . Fields are measured by Zeeman splitting and the Chandrasekhar–Fermi method.

If turbulence decays in one crossing time, why are clouds still turbulent?

Because something keeps stirring them. Simulations show supersonic turbulence — with or without magnetic fields — loses roughly half its kinetic energy per crossing time, decaying as a power law rather than persisting. A cloud that lives many crossing times must therefore be continuously driven. The leading candidates are supernovae and expanding H II regions on large scales, galactic shear, accretion onto the cloud, and outflows and radiation from the stars it is already forming. Disentangling which dominates, and in what mix of compressive and solenoidal forcing, is an open problem, because that mix sets the density PDF and hence the star-formation rate.