Turbulence, rain drops and the l**1/2 number density law and downscaling
Résumé
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 lc is the critical scale where the Stokes number (= drop inertial time/turbulent eddy time) St l is unity. For ?ve storms, we found lc in the range 45 -75 cm with the corresponding dissipation scale St? in the range 200 - 300. Since the mean interdrop distance was signi?cantly smaller ( 10cm) than lc we infer that rain consists of patches whose mean liquid water content is determined by turbulence with each patch being statistically homogeneous. For l higher than lc, we have St l lower than 1 and due to the observed statistical homogeneity for l lower than lc, we argue that we can use Maxeys relations between drop and wind velocities at coarse grained resolution lc. From this, we derive equations for the number and mass densities (n, ?) and their variance ?uxes (?, ?). By showing that ?, is dissipated at small scales (with l?,diss lc) and n over a wide range, we conclude that ? should indeed follow Corrsin-Obukhov k -5/3 spectra but that n should instead follow a k -2 spectrum corresponding to ?uctuations scaling as ?? l 1/3 , ?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 ?n l 1/2 number density law is quite new. The key difference between the ??, ?n laws is the fact that the microphysics (coalescence, breakup) conserves drop mass, but not numbers. This implies that the time scale for the transfer of the density variance ?ux ? is determined by the strongly scale dependent turbulent velocity whereas the time scale for the transfer of the number variance ?ux ? 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 n, ? ?elds which are the 0 th , 1 st moments of the the drop size distribution, they constrain the microphysics: dimensional analysis shows that the cumulative probability distribution of nondimensional drop mass (Mn/?) should be a universal function dependent only on scale; we con?rm 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 modeling rain. We discuss the implications of this for the rain rate statistics including a simpli?ed model which can explain the observed rain rate spectra. Finally, we discuss the implications for rainfall downscaling.