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# On Estimating Hail Frequency and Hailfall Area

Alexis B. Long

## Abstract

Probabilistic and statistical concepts are used to examine how the number of hail observing sites within a region affects the accuracy of estimates of 1) the mean point frequency of hail within the region, 2) the overall regional frequency of hail, and 3) the area covered by individual hailfalls. A practically useful relationship ν̄ = /A is first derived to connect the mean point frequency ν̄ of hail within a region to the regional hail frequency N, to the mean area ā of the individual hailfalls and to the area A of the region. The error in estimating the mean frequency ν̄ is found to increase as n −1/2, where n is the number of sites placed within a region. If within a region there are proportionally more large hailstorms or if most of the area covered by hail is commonly due to a few large hailstorms, then fewer sites will be needed to estimate the mean point hail frequency. Of the 16 hailfalls detected by the 660 km2 1976 National Hail Research Experiment (NHRE) network of 603 hailpad sites, it is found, using a simple probabilistic expression, that 12 of the hailfalls still would have been detected using only 50 sites. The smaller hailfalls would have been the first to go undetected. There are diminishing returns in fielding sufficient instruments to detect all, or almost all, of the hailfalls in a region. For hailfalls with the lognormal distribution of areas observed by NHRE, the number of instruments needed increases exponentially with the number of hailfalls to be detected. A formula is derived for a correction to be made to the observed regional frequency of hailfalls of a given size so as to obtain the true frequency. For random networks the coefficient of variation of an estimate of the area of a hailfall is proportional to n −1/2. For a hailfall less than one-fifth as large as the instrumented region in which it lies, the coefficient of variation of an estimate of its area approximately equals nh −1/2, where nh is the expected number of sites within the hailfall.

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# Solutions to the Droplet Collection Equation for Polynomial Kernels

Alexis B. Long

## Abstract

Numerical solutions to the droplet collection equation, using certain polynomial approximations to the gravitational collection kernel, are examined to learn whether they usefully describe the evolution of a cloud droplet size distribution. The results for typical continental and maritime clouds show that the distribution is closely described if the kernel is replaced by
9x2y2R3xyR
or by
10x2R3xR
where R is the radius of the larger droplet, x its volume in cubic centimeters, and y the volume of the smaller droplet.

From the standpoint of including collision and coalescence of droplets in multi-dimensional cloud models an analytic solution to the collection equation is desirable. An attempt should be made to find such solutions based upon either of the above approximations. If these cannot be found because of the piecewise nature of the approximations, then solutions based on the portions for R≤50 μm would still describe the first few hundred seconds of droplet growth. A comparatively poor description of the droplet distribution comes from the most physically realistic analytic solution presently existing, based on the kernel approximation B(x+y)+Cxy.

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# Australian Winter Storms Experiment (AWSE) I: Supercooled Liquid Water and Precipitation-Enhancement Opportunities

Alexis B. Long
and
Arlen W. Huggins

## Abstract

Some results of the first (1988) Australian Winter Storms Experiment are described. The results shed light on precipitation-enhancement opportunities in winter cyclonic storms interacting with the Great Dividing Range of southeast Australia. The results come from analysis of supercooled liquid water amounts provided by a dual-wavelength microwave radiometer, atmospheric structure from Omegasondes, and precipitation amounts from a large number of tipping-bucket gauges. With these data it is possible to calculate and compare two of the terms in a condensed-phase water budget over a cloud-seeding target area in the Great Dividing Range. The two terms are the horizontal flux of supercooled liquid cloud water entering the budget volume and the vertical precipitation flux at ground level out of the volume. The budget terms have implications for the amount of extra precipitation that may result from seeding. It is found that the amount depends on the frontal or postfrontal stage of activity in the target area and on the wind direction with respect to the mountainous terrain.

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# Australian Winter Mountain Storm Clouds: Precipitation Augmentation Potential

Alexis B. Long
and
Elizabeth J. Carter

## Abstract

Two Australian winter mountain storm field research projects were conducted by the Commonwealth Scientific and Industrial Research Organisation Division of Atmospheric Research and the Desert Research Institute Atmospheric Sciences Center in the austral winters of 1988 and 1990. These projects gained information about winter storms in support of the ongoing Melbourne Water randomized cloud seeding experiment aimed at increasing runoff into Melbourne's main water supply, the Thomson Reservoir. This paper discusses some of the 1988 instrumentation data. One variable of interest is the precipitation augmentation potential π. It is the difference between (a) the horizontal supercooled liquid water flux in the clouds crossing the mountains and (b) the vertical precipitation flux at the surface from the clouds. These fluxes are based on calculations of supercooled liquid water depth in clouds with a microwave radiometer, Omegasonde wind velocity, and rates of precipitation from gauges. It was found that π varies systematically during a winter storm. The greatest potential occurs in the post-cold-frontal stage of a storm when the cloud-top temperature is warm and about −12°C and the wind direction of 240° is approximately orthogonal to the main southwest face of the predominant orographic feature, Baw Baw Plateau, of the study area. The potential is significantly less during the prefrontal and frontal stages, with cloud-top temperatures of about −35°C and a wind direction of about 3O0° parallel to the Baw Baw Plateau. The results show that cloud seeding would have the greatest benefit in the postfrontal stage.

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# On the Evaluation of the Collection Kernel for the Coalescence of Water Droplets

Alexis B. Long
and
Michael J. Manton

## Abstract

Two anomalies are described which arise in the kernel for stochastic droplet collection when it is specified by the formula of Scott and Chen for the linear collision efficiency y(R,r) and by the formula of Wobus et al. for the droplet terminal velocity V(R). It is pointed out that if accurate values for y(R,r) are to be obtained for a given droplet pair by interpolation using data for specific droplet pairs, then for large droplet radii (R<30 μm) it is desirable that these data he tabulated for 2-μm intervals of R. It is shown that if the difference in terminal velocities of two droplets is computed from a formula approximating V(R) and composed of various functions V*(R) applicable over adjoining domains of R, then it is necessary that these functions be constructed so that the formula and its derivatives, at least up to second order, are everywhere continuous. An improved formula for V(R) satisfying this criterion is described.

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# Investigations of a Winter Mountain Storm in Utah. Part I: Synoptic Analyses, Mesoscale Kinematics, and Water Release Rates

Alexis B. Long
,
Arlen W. Huggins
, and
Bernard A. Campistron

## Abstract

A winter storm passing across the north–south-orientated Tushar Mountains in southwest Utah is investigated in this multipart paper. This Part I describes the evolving synoptic pattern, mesoscale kinematics, and calculated water release rates (condensation or deposition) in clouds over the western upstope part of the mountains. Horizontal mesoscale kinematic variables come from direct application of Volume Velocity Processing to single C-band Doppler radar data. Water release rates are computed from updrafts derived from the radar data and from the vertical gradient of saturation mixing ratio obtained from soundings.

In Stage I of the storm altostratus was present on the leading side of a long-wave trough. Weak updrafts occurred only at the higher altitudes within the clouds where there was convergence and large-scale synoptically forced lift. Downdrafts as great as −0.6 m s−1 occurred in the lower parts of the cloud where there was divergence. The downdrafts were induced in part by sublimation cooling of solid (ice) precipitation falling from the altostatus. Only virga was observed and the radar echoes did not reach the surface.

Stage II was initially dominated by passage of a short-wave aloft. Drier air associated with the short-wave led to complete evaporation of the altostratus of Stage I. The lower parts of this cloud (≤4.5 km MSL) eventually redeveloped into altocumulus.

Later in Stage II the wind veered more perpendicular to the mountains. Simultaneously, convergence developed in the lower 900–1200 m of the atmosphere, and mesoscale updrafts of 0.1–0.2 in m s−1 were calculated. Maxima in the water release rate were associated with the updrafts.

During Stage III a passing cold front influenced the kinematics and cloud and precipitation. From prior to frontal passage to a few hours afterward the wind beneath the frontal surface veered from southwesterly to northerly. There was strong convergence at low altitudes just upwind of the Tushar Mountains. It was accompanied by strong, deep mesoscale updrafts extending from near the ground up through the frontal surface and by water release maxima.

The storm changed character after the wind at low altitudes had veered to northerly and had become parallel to the Tushar Mountains. Convergence maxima continued to be present beneath the frontal surface but weaker. They preceded by ∼0.5 h maxima in the convergence above the frontal surface. Associated with these paired convergence features were updraft maxima located above the frontal surface. Water release rates were generally lower than earlier in Stage III. The decrease was greatest at low altitudes beneath the frontal surface where the wind had veered to northerly, where there was little uplift by the Tushar Mountains, and where updrafts were weak. Above the frontal surface the decrease in water release rate was not as great inasmuch as lift by the frontal surface was still occurring.

The storm dissipated in Stage IV. The axis of the longwave trough passed through the area, winds at higher altitudes beneath the frontal surface veered more northerly, and there was substantial drying at all altitudes above and below the frontal surface. The winds beneath the frontal surface were divergent, indicative of subsidence, and mesoscale downdrafts were present.

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# The Hailpad: Materials, Data Reduction and Calibration

Alexis B. Long
,
Richard J. Matson
, and
Edwin L. Crow

## Abstract

This paper reports on work carried out in the National Hail Research Experiment (NHRE) on hailpad materials, on procedures for reducing hailpad data, and on hailpad calibration. A recommendation is made for a pad constructed of 2.5 cm thick type-SI Styrofoam (manufactured by Dow Chemical USA) and sprayed with a 25–50 μm coating of white latex paint for protection from the deteriorating effects of sun-light. Calibration of the hailpad provides a relation between the minor axis of a dent in the pad and the dimensions of the stone producing the dent. It is recommended that measurements of the minor axis be categorized in size intervals no wider than 4 mm.

The NHRE laboratory technique for calibrating hailpads involves simulating a hailstone impact by dropping a steel sphere onto a pad from a height such that the impact kinetic energy achieved by the sphere equals that of a hailstone of equal diameter falling onto the pad in an environment with known horizontal wind. The pad is tilted to preserve the stone impact angle found in nature. A second-degree polynomial in sphere diameter D satisfactorily describes the calibration relation between D and the dent minor axis. Application of the calibration relation developed for the particular case of no wind to hailpads which have been hit by hail falling in a wind leads to an overestimate of hailstone diameter of approximately 0.5–1% per meter per second of wind speed. This effect of the wind is about twice as large as that found by others.

A theoretical expression is developed that explicitly relates the minor axis of a dent produced by a sphere to the diameter of the sphere. Two controlling parameters in this expression are the impact kinetic energy of the sphere and a factor p, with dimensions of pressure, which quantitatively embodies the response of a pad to a sphere impact. The effect of variations in p on the sphere diameter derived from dent minor axis and information supplied by Dow Chemical USA on possible variability in the compressive modulus of Styrofoam between manufacturing batches together suggest that the user of hailpads obtains a one time all the foam he may need for his work.

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# Investigations of a Winter Mountain Storm in Utah. Part III: Single-Doppler Radar Measurements of Turbulence

Bernard Campistron
,
Arlen W. Huggins
, and
Alexis B. Long

## Abstract

This Part III of a multipart paper deals with the analysis of turbulent motion in a winter storm, which occurred over the mountains of southwest Utah. The storm was documented with a long duration single Doppler radar dataset (∼21 h) comprised of volume scan observations acquired at 10-min intervals. Turbulence parameters were determined using a new technique of volume processing of single Doppler radar data.

Physical analysis of turbulence is restricted to three particular storm regions: a prefrontal region far removed from a cold frontal discontinuity, a frontal zone aloft, and a low layer in the post-frontal region where a long lasting (∼6 h) wind-maximum existed. The prefrontal period showed enhancement of turbulent parameters near 2.6 km height, apparently due to disturbed flow caused by an upwind mountain range. Turbulence parameters in this prefrontal region showed good agreement with K-mixing length theory. Within the frontal zone most turbulence parameters reached peak values, but were generally less than orographically induced turbulence values in the prefrontal period.

Turbulence in the low-level postfrontal period experienced periodic oscillations consistent with precipitation and kinematic variables described in Parts I and II, and associated with mesoscale precipitation bands. Acceleration of the valley-parallel wind component was apparent in prefrontal and postfrontal periods and was related to the specific valley configuration through a Venturi effect.

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# A Method of Retrieving Turbulence Parameters from Volume Processing of Single-Doppler Radar Measurements

Bernard Campistron
,
Alexis B. Long
, and
Arlen W. Huggins

## Abstract

In previous work the derivation of turbulence parameters from single-Doppler radar observations was performed with data acquired along a horizontal circle. Here the technique is extended to all the radar data within a horizontal cylindrical slice of finite depth using the same basic assumptions of linearity of the mean wind field and horizontal homogeneity of the turbulence. The method allows the extraction of the six Reynolds stress components, together with their vertical derivatives, and the turbulent fluxes of a scalar quantity deduced from the reflectivity data.

Experimental data were used for the performance evaluation of the methodology. A simple testing procedure was carried out to remove erroneous results. The statistical uncertainty in the measured Reynolds stress terms was found to be about 0.05 m2 s−2, except for the variance of the vertical component, which was poorly retrieved because of an absence of data at high elevation angles. Calculations showed that contamination of the vertical momentum flux measurements by the scatterer fall speed was negligible. An analysis of the response function of the technique to the atmospheric scales tended to show that the diameter of the processing slices corresponded to the largest turbulent scale dimension involved in the measured turbulence quantities.

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# Some Examples of Rapidly Growing Oceanic Cumulonimbus Clouds

Helmut K. Weickmann
,
Alexis B. Long
, and
L. Ray Hoxit

## Abstract

Infrared satellite photographs of the tropical oceanic regions within and around the GATE A/B array point to the existence of large, rapidly growing, cumulonimbus clouds. The region along 10°N experiences the greatest frequency of these storms. Also a pronounced diurnal variation is found in the times of initial development with maximum frequencies near midnight. In most cases, this anvil cloud grows to an areal extent ∼7000 km2 in about 4 h, then dissipates in another 3 h. The velocity divergence in the anvil is 1−3×10−4s−1 and the mass outflow is 100–200 kton s−1. These storms account for some of the difference in diurnal variation of high cloudiness that is observed between tropical oceanic and continental regions. A large number of rapidly growing cumulonimbus clouds was observed on 10 August 1974 in connection with a disturbance in the tropical easterlies that ultimately developed into Tropical Storm Alma.

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