Turbulence: a pattern that repeats over many decades of scale, and the two sizes where it stops #19

Question

A turbulent flow looks statistically the same at every size between the scale where energy is stirred in and the scale where it turns into heat. Can a small model predict both ends of that range from first principles, and reproduce the measured slopes in between, including where they deviate?

• Lower end in air and water: the Kolmogorov scale η = (ν³/ε)^¼, about a millimetre in the atmosphere and upper ocean.
• Lower end in the solar wind: the ion scale. The proton inertial length is about 100 km at 1 AU, which by Taylor's hypothesis puts the spectral break near 0.6 Hz.
• Lower end in molecular clouds: the sonic scale, where the velocity spread falls to the sound speed, about 0.07 pc from σ = 0.72 S^0.5 km/s at 10 K.

Why it matters

Turbulence is the best-measured case of real self-similarity with both cutoffs physically understood. The interstellar electron-density spectrum follows one Kolmogorov-like power law over many decades (the "big power law in the sky", Armstrong, Rickett and Spangler, ApJ 443, 209 (1995)). The repetition is not perfect, and the model can show where:

• intermittency (She and Leveque, PRL 72, 336 (1994));
• a change of cascade in molecular clouds (Larson's σ ∝ L^0.38, MNRAS 194, 809 (1981), against the supersonic 0.5 of Solomon et al. 1987);
• a 2D-like k⁻³ range in the atmosphere above ~500 km (Nastrom and Gage, J. Atmos. Sci. 42, 950 (1985)).

It also meets the quantum thread: in superfluid helium the classical cascade hands over to quantised vortices, with circulation h/m.

First steps

• turbulence/model.js: η and L/η ~ Re^¾; the solar-wind break from the ion inertial length; the molecular-cloud sonic scale.
• Checks against published numbers: the exact 4/5 law; the Kolmogorov constant (~1.5); She–Leveque exponents against measured ones; the solar-wind break at 0.1–1 Hz with a kinetic-range slope near −2.8 (Bruno and Carbone, Living Rev. Solar Phys. 10, 2 (2013)); the sonic scale against the ~0.1 pc coherent cores and filament widths. The numbers in this issue are from recall in a session whose literature search was exhausted, so verify them first.
• Larson's 0.38 against 0.5: which cascade the cloud data favour, and where the analogy with lab turbulence breaks (compressible, magnetised, self-gravitating).
• Cloud edges: their perimeter-area dimension (~1.36, Falgarone et al. 1991) does not follow simply from a Kolmogorov spectrum (a fractional-Brownian estimate gives 7/6). Test whether the agreement people cite is real or a coincidence.