Turbulence, raindrops and the<i>l</i>^1/2number density law

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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