Turbulence, raindrops and the<i>l</i><sup>1/2</sup>number density law
Bibliographic record
Abstract
Using a unique data set of three-dimensional drop positions and masses (the HYDROP experiment), we show that the distribution of liquid water in rain displays a sharp transition between large scales which follow a passive scalar-like Corrsin–Obukhov ( k -5/3 ) spectrum and a small-scale statistically homogeneous white noise regime. We argue that the transition scale l c is the critical scale where the mean Stokes number (= drop inertial time/turbulent eddy time) St l is unity. For five storms, we found l c in the range 45–75 cm with the corresponding dissipation scale St η in the range 200–300. Since the mean interdrop distance was significantly smaller (≈ 10 cm) than l c we infer that rain consists of 'patches' whose mean liquid water content is determined by turbulence with each patch being statistically homogeneous. For l > l c , we have St l <1 and due to the observed statistical homogeneity for l < l c , we argue that we can use Maxey's relations between drop and wind velocities at coarse grained resolution l c . From this, we derive equations for the number and mass densities ( n and ρ) and their variance fluxes (ψ and χ). By showing that χ is dissipated at small scales (with l ρ, diss ≈ l c ) and ψ over a wide range, we conclude that ρ should indeed follow Corrsin–Obukhov k -5/3 spectra but that n should instead follow a k -2 spectrum corresponding to fluctuations scaling as Δρ∝ l 1/3 and Δ n ∝ l 1/2 . While the Corrsin–Obukhov law has never been observed in rain before, its discovery is perhaps not surprising; in contrast the Δ n ≈ l 1/2 number density law is quite new. The key difference between the Δρ, Δ n laws is the fact that the microphysics (coalescence, breakup) conserves drop mass, but not numbers of particles. This implies that the timescale for the transfer of the density variance flux χ is determined by the strongly scale-dependent turbulent velocity whereas the timescale for the transfer of the number variance flux ψ is determined by the weakly scale-dependent drop coalescence speed. We argue that the l 1/2 law may also hold (although in a slightly different form) for cloud drops. Because they are consequences of symmetries, we expect the l 1/3 , l 1/2 laws to be robust. Since the large-scale turbulence determines the n and ρ fields which are the 0th and 1st moments of the drop-size distribution, they constrain the microphysics: dimensional analysis shows that the cumulative probability distribution of nondimensional drop mass should be a universal function dependent only on scale; we confirm this empirically. The combination of number and mass density laws can be used to develop stochastic compound multifractal Poisson processes which are useful new tools for studying and modelling rain. We discuss the implications of this for the rain rate statistics including a simplified model, which can explain the observed rain rate spectra.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".