Clustering and collision of inertial particles in random velocity fields

The influence of clustering on the collision rate of inertial particles in a smooth random velocity field, mimicking the smaller scales of a turbulent flow, is analyzed. For small values of the the ratio between the relaxation time of the particle velocity and the characteristic time of the field, the effect of clusters is to make more energetic collisions less likely. The result is independent of the flow dimensionality and is due only to the origin of collisions in the process of caustic formation.Comment: 4 pages, 3 figures, revtex

Drizzle rates versus cloud depths for marine stratocumuli

Marine stratocumuli make a major contribution to Earth’s radiation budget. Drizzle in such clouds can greatly affect their albedo, lifetime and fractional coverage, so drizzle rate prediction is important. Here we examine a question: does a drizzle rate (R) depend on cloud depth (H) and/or drop number concentration n in a simple way? This question was raised empirically in several recent publications and an approximate H3/n dependence was observed. Here we suggest a simple explanation for H3 scaling from viewing the drizzle rate as a sedimenting volume fraction ( f ) of water drops (radius r) in air, i.e. R = f u(r ), where u is the fall speed of droplets at the cloud base. Both R and u have units of speed. In our picture, drizzle drops begin from condensation growth on the way up and continue with accretion on the way down. The ascent contributes H ( f ∝ H) and the descent H2 (u ∝ r ∝ f H) to the drizzle rate. A more precise scaling formula is also derived and may serve as a guide for parameterization in global climate models. The number concentration dependence is also discussed and a plausibility argument is given for the observed n−1 dependence of the drizzle rate. Our results suggest that deeper stratocumuli have shorter washout times

Simple approximations for condensational growth

A simple geometric argument relating to the liquid water content of clouds is given. The phase relaxation time and the nature of the quasi-steady approximation for the diffusional growth of cloud drops are elucidated directly in terms of water vapor concentration. Spatial gradients of vapor concentration, inherent in the notion of quasi-steady growth, are discussed and we argue for an occasional reversal of the traditional point of view: rather than a drop growing in response to a given supersaturation, the observed values of the supersaturation in clouds are the result of a vapor field adjusting to droplet growth. Our perspective is illustrated by comparing the exponential decay of condensation trails with a quasi-steady regime of cirrus clouds. The role of aerosol loading in decreasing relaxation times and increasing the rate of growth of the liquid water content is also discussed

Secular changes in atmospheric turbidity over Iraq and a possible link to military activity

We examine satellite-derived aerosol optical depth (AOD) data during the period 2000-2018 over the Middle East to evaluate the contribution of anthropogenic pollution. We focus on Iraq, where US troops were present for nearly nine years. We begin with a plausibility argument linking anthropogenic influence and AOD signature. We then calculate the percent change in AOD every two years. To pinpoint the causes for changes in AOD on a spatial basis, we distinguish between synoptically calm periods and those with vigorous synoptic activity. This was done on high-resolution 10 km AOD retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor (Terra satellite). We found spatiotemporal variability in the intensity of the AOD and its standard deviation along the dust-storm corridor during three studied periods: before Operation Iraqi Freedom (OIF) (1 March 2000-19 March 2003), during OIF (20 March 2003-1 September 2010), and Operation New Dawn (OND; 1 September 2010-18 December 2011), and after the US troops\u27 withdrawal (19 December 2011-31 December 2018). Pixels of military camps and bases, major roads and areas of conflict, and their corresponding AOD values, were selected to study possible effects. We found that winter, with its higher frequency of days with synoptically calm conditions compared to spring and summer, was the best season to quantitatively estimate the impact of these ground-based sources. Surprisingly, an anthropogenic impact on the AOD signature was also visible during vigorous synoptic activity. Meteorological conditions that favor detection of these effects using space imagery are discussed, where the effects are more salient than in surrounding regions with similar meteorological conditions. This exceeds expectations when considering synoptic variations alone

What is raindrop size distribution?

It is commonly understood that the number of drops that one happens to measure as a function of diameter in some sample represents the drop size distribution. However, recent observations show that rain is “patchy” suggesting that such a seemingly “obvious” definition is incomplete. That is, rain consists of patches of elementary drop size distributions over a range of different scales. All measured drop size distributions, then, are statistical mixtures of these patches. Moreover, it is shown that the interpretation of the measured distribution depends upon whether the rain is statistically homogeneous or not. It is argued and demonstrated using Monte Carlo simulations that in statistically homogeneous rain, as the number of patches included increases, the observed spectrum of drop sizes approaches a “steady” distribution. On the other hand, it is argued and demonstrated using video disdrometer data that in statistically inhomogeneous rain, there is no such steady distribution. Rather as long as one keeps measuring, the drop size distribution continues to change. What is observed, then, depends on when one chooses to stop adding measurements. Consequently, the distributions measured in statistically inhomogeneous rain are statisticalentities of mean drop concentrations best suited to statistical interpretations. In contrast, steady distributions in statistically homogeneous rain are more amenable to deterministic interpretations since they depend upon factors independent of the measurement process. These findings have implications addressed in two additional questions, namely, Are computer-created virtual drop size distributions really the same as those observed? What is the appropriate drop size distribution when several measurements used in an algorithm for rain estimations are made at different resolutions

Non-Rayleigh signal statistics in clustered statistically homogeneous rain

As the sample volume of a remote sensing instrument moves through sufficiently variable conditions, recent work shows that the amplitudes and associated intensities can deviate significantly at times from expectations based on Rayleigh signal statistics because fluctuations in the number of scatterers leads to a doubly stochastic measurement process. While non-Rayleigh deviations yield average biases for both logarithmic and linear detectors, perhaps of greater importance is the enhancement of the variance of the bias distribution for square law detectors. In this work the authors explore the potential existence of non-Rayleigh effects even in the statistically homogeneous rain when fluctuations in the number of scatterers should be much less than for the inhomogeneous conditions used in earlier studies. Moreover, in contrast to previous work, recent advances now permit the simulation of correlated rainfall structures having the statistical characteristics of natural rain such as clustering intensity (ℵ) and coherence length (χ) consistent with observations. The primary objective of this work, then, is to clarify how ℵ, χ, and the geometric parameters characteristic of remote sensing observations such as the distance over which an estimate is made (L), the beamwidth (B), and the spatial displacement between successive independent samples (Δ) affect non-Rayleigh signals statistics in statistically homogeneous rain. This work shows that non-Rayleigh effects can appear whenever Δ ⩽ χ ⩽ L. Moreover, the magnitudes of the non-Rayleigh deviations increase as ℵ and Δ/B increase. Although non-Rayleigh effects can be detected by monitoring of the signals, keeping both Δ/B and L as small as possible while increasing sample independence using chirp or signal whitening techniques, for example, should help to minimize non-Rayleigh effects for radars even in statistically inhomogeneous rain

When is rain steady?

By definition, steady rain should have a nearly constant rainfall rate. Thus far, however, the criteria for determining when rain is steady remain qualitative and arbitrary. The authors suggest a definition for steadiness that can be used to quantify the elusive notion of natural variability. In particular, the logical criteria for steadiness imply statistical stationarity and lack of correlation between raindrops in neighboring volumes, requirements identical to those for the drops being distributed according to a Poisson process at all scales. Hence, steady rain is Poissonian. Explicit equations for the variance of the rainfall rates are developed. They show that, in general, raindrop clustering enhances the variance beyond that for Poissonian rain (). It is also demonstrated by using observations that this enhancement is augmented further when the rain is statistically nonstationary. Identifying steady rain is important. To be specific, because steady rain is statistically stationary, the drop size distributions have physical, deterministic meanings independent of the measurement process. Observables such as the radar reflectivity factor and the rainfall rate are then steady and linearly related also. Techniques for determining when rain is steady are discussed. The ratio / is proposed as a useful quantitative measure of the steadiness of the rain. It is also shown that an estimate of the minimum possible standard deviation for steady rain is / where and are the mean rain rate and average number of drops per sample, respectively. Examples using video-disdrometer data are also presented

Non-Rayleigh signal statistics caused by relative motion during measurements

In order to reduce fluctuations, remote sensing devices such as radars and radiometers typically sample many times before forming an estimate. When mean values are stationary during this sampling period, the fluctuations in the amplitudes and intensities obey the same probability density functions (pdf) as those for each sample contributing to the estimate. However, it is shown in this work that when mean values change from sample to sample (i.e., pulse to pulse for most radars), the pdf\u27s of the amplitudes and intensities differ from those corresponding to the samples. Such changes can be inherent to the scatterers as, for example, the scatter of microwaves from an ocean surface, or they can be induced by factors such as antenna motion across gradients. With respect to meteorological radars, it is routinely argued that the central limit theorem leads inexorably to zero-mean Gaussian distributions of the two components of the electric field phasor backscattered from precipitation because of the large number of independent scatterers in the sampling volume. Consequently, the net amplitudes and intensities obey Rayleigh and exponential probability density distributions, respectively. While apparently true for each pulse (sample) even when the reflectivity across the beam is not uniform, the authors show that, in general, the underlying statistics of the amplitudes and intensities are no longer Rayleigh nor exponential. This occurs because the number of scatterers and intensities change from sample to sample as, for example, when a radar beam moves while the mean intensity is changing. Consequently, non-Rayleigh statistics and deviations from Gaussian distributions are probably much more common than previously appreciated. A statistical model is developed and confirmed from detailed Monte Carlo drop simulators of a radar sampling as the beam moves through a cloud. Theory and these model simulations show that the resultant pdf\u27s of the amplitude and intensity are mixtures of the pdf\u27s from each sample contributing to the estimate. This mixture of pdf\u27s also produces increased variance. Because of the general nature of these findings, it is likely that the effects of sampling through changing conditions (namely, biases and increased variances) probably also apply to many other types of remote sensing instruments including those using square-law detectors

Direct observations of coherent backscatter of radar waves in precipitation

In previous work, it was argued that a source of radar coherent scatter occurs in the direction perpendicular to the direction of wave propagation because of the presence of grids of enhanced particle concentrations with spatial periodicities in resonance with the radar wavelength. While convincing, the evidence thus far has been indirect. In this work the authors now present direct observations of radar coherent backscattered signals in precipitation in the direction of wave propagation. The theory is developed for the cross-correlation function of the complex amplitudes in the direction of propagation calculated for nearest neighbor range bins. Data are analyzed in snow and in rain. The results agree with the earlier conclusions in the previous work, namely that coherent scatter occurs in both rain and snow, that it is larger in snow than it is in rain, and that it can be significant at times

Partially coherent backscatter in radar observations of precipitation

Classical radar theory only considers incoherent backscatter from precipitation. Can precipitation generate coherent scatter as well? Until now, the accepted answer has been no, because hydrometeors are distributed sparsely in space (relative to radar wavelength) so that the continuum assumption used to explain coherent scatter in clear air and clouds does not hold. In this work, a theory for a different mechanism is presented. The apparent existence of the proposed mechanism is then illustrated in both rain and snow. A new power spectrum Z( f ), the Fourier transform of the time series of the radar backscattered reflectivities, reveals statistically significant frequencies f of periodic components that cannot be ascribed to incoherent scatter. It is shown that removing those significant fs from Z( f ) at lower frequencies greatly reduces the temporal coherency. These lower frequencies, then, are associated with the increased temporal coherency. It is also shown that these fs are also directly linked to the Doppler spectral peaks through integer multiples of one-half the radar wavelength, characteristic of Bragg scatter. Thus, the enhanced temporal coherency is directly related to the presence of coherent scatter in agreement with theory. Moreover, the normalized backscattered power spectrum Z( f ) permits the estimation of the fractional coherent power contribution to the total power, even for an incoherent radar. Analyses of approximately 26 000 one-second Z( f ) in both rain and snow reveal that the coherent scatter is pervasive in these data. These findings present a challenge to the usual assumption that the scatter of radar waves from precipitation is always incoherent and to interpretations of backscattered power based on this assumption