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

Obtaining the drop size distribution

his document is a supplement to “Fluctuations and Luck in Droplet Growth by Coalescence,” by Alexander B. Kostinski and RaymondA. Shaw (Bull. Amer. Meteor. Soc.,86, 235–244) • ©2005 American Meteorological Societ

Spatial patterns of record-setting temperatures

We employ record-breaking statistics to study spatial correlations of record-setting terrestrial surface temperatures. To that end, a simple diagnostic tool is devised, reminiscent of a pair-correlation function. Data analysis reveals that while during the hottest years, record-breaking temperatures arrive in “heat waves”, extending throughout almost the entire continental United States, this is not so for all years, not even recently. Record-breaking temperatures generally exhibit spatial patterns and variability quite different from those of the mean temperatures

Fluctuations and luck in droplet growth by coalescence

After the initial rapid growth by condensation, further growth of a cloud droplet is punctuated by coalescence events. Such a growth process is essentially stochastic. Yet, computational approaches to this problem dominate and transparent quantitative theory remains elusive. The stochastic coalescence problem is revisited and it is shown, via simple back-of-the-envelope results, that regardless of the initial size, the fastest one-in-a-million droplets, required for warm rain initiation, grow about 10 times faster than the average droplet. While approximate, the development presented herein is based on a realistic expression for the rate of coalescence. The results place a lower bound on the relative velocity of neighboring droplets, necessary for warm rain initiation. Such velocity differences may arise from a variety of physical mechanisms. As an example, turbulent shear is considered and it is argued that even in the most pessimistic case of a cloud composed of single-sized droplets, rain can still form in 30 min under realistic conditions. More importantly, this conclusion is reached without having to appeal to giant nuclei or droplet clustering, only occasional “fast eddies.” This is so because, combined with the factor of 10 accelerated growth of the one-in-a-million fastest droplets, the traditional turbulent energy cascade provides sufficient microshear at interdroplet scales to initiate warm rain in cumulus clouds within the observed times of about 30 min. The simple arguments presented here are readily generalized for a variety of time scales, drizzle production, and other coagulation processes

Evolution and distribution of record-breaking high and low monthly mean temperatures

The ratio of record highs to record lows is examined with respect to extent of time series for monthly mean temperatures within the continental United States for 1900–2006. In counting the number of records that occur in a single year, the authors find a ratio greater than unity in 2006, increasing nearly monotonically as the time series increases in length via a variable first year over 1900–76. For example, in 2006, the ratio of record highs to record lows ≈ 13:1 with 1950 as the first year and ≈ 25:1 with 1900 as the first year; both ratios are an order of magnitude greater than 3σ for stationary simulations. This indicates a warming trend. It is also found that records are more sensitive to trends in time series of monthly averages than in time series of corresponding daily values. When the last year (1920–2006, starting in 1900) is varied, it is found that the ratio of record highs to record lows is strongly correlated with the ensemble mean temperature. Correlation coefficients are 0.76 and 0.82 for 1900–2006 and 1950–2006, respectively; 3σ = 0.3 for pairs of uncorrelated stationary time series. Similar values are found for globally distributed time series: 0.87 and 0.92 for 1900–2006 and 1950–2006, respectively. The ratios evolve differently, however: global ratios increase throughout (1920–2006) whereas continental U.S. ratios decrease from about 1940 to 1970. Last, the geographical and seasonal distributions of trends are considered by summing records over time rather than ensemble. In the continental United States, the greatest excess of record highs occurs in February (≈2:1) and the greatest excess of record lows occurs in October (≈2:3). In addition, ratios are pronounced in certain regions: in February in the Midwest the ratio ≈ 5:2, and in October in the Southeast the ratio ≈ 1:2

Record-breaking statistics for random walks in the presence of measurement error and noise

We address the question of distance record-setting by a random walker in the presence of measurement error, $\delta$, and additive noise, $\gamma$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $\sim n^{1/2}$ for large $n$ for all jump distributions, including L\'evy flights, and for all $\delta$ and $\gamma$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $\delta$ and $\gamma$. In the absence of noise ($\gamma=0$), the pre-factor $S(\delta)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $\delta$ whereas, in case of perfect measurement $(\delta=0)$, the corresponding pre-factor $T(\gamma)$ increases with $\gamma$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions