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- Author or Editor: R. C. Srivastava x

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## Abstract

For an exponential drop size distribution, the mean Doppler velocity at vertical incidence, after correction for vertical air motion, depends only on the slope of the distribution. The intercept of the distribution can then be deduced from the radar reflectivity and the slope and the intercept can be used to compute the rainwater content and the rainfall rate. The necessity of correcting for the vertical air motion limits the method to stratiform rain where it may be obtained by VAD methods or may be small enough to be neglected. Two examples of the use of the method are presented.

## Abstract

For an exponential drop size distribution, the mean Doppler velocity at vertical incidence, after correction for vertical air motion, depends only on the slope of the distribution. The intercept of the distribution can then be deduced from the radar reflectivity and the slope and the intercept can be used to compute the rainwater content and the rainfall rate. The necessity of correcting for the vertical air motion limits the method to stratiform rain where it may be obtained by VAD methods or may be small enough to be neglected. Two examples of the use of the method are presented.

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## Abstract

Numerical computations of the modification of raindrop size distribution by coalescence are presented. The initial distribution is defined by giving the drop concentrations at a discrete set of drop sizes of which the volumes are integral multiples of an elementary volume; in this way, the drop sizes produced by coalescence remain confined to the discrete set.

It is found that an initial exponential distribution of the Marshall-Palmer type undergoes comparatively little change. On the other hand, much narrower initial distributions, consisting of drops of two sizes only, are found to undergo considerable change and tend toward an exponential distribution. The possible role of coalescence in generating observed raindrop size distributions from narrow initial distributions is discussed.

## Abstract

Numerical computations of the modification of raindrop size distribution by coalescence are presented. The initial distribution is defined by giving the drop concentrations at a discrete set of drop sizes of which the volumes are integral multiples of an elementary volume; in this way, the drop sizes produced by coalescence remain confined to the discrete set.

It is found that an initial exponential distribution of the Marshall-Palmer type undergoes comparatively little change. On the other hand, much narrower initial distributions, consisting of drops of two sizes only, are found to undergo considerable change and tend toward an exponential distribution. The possible role of coalescence in generating observed raindrop size distributions from narrow initial distributions is discussed.

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## Abstract

The effect of precipitation on the properties of a one-dimensional non-entraining cumulus is studied by numerically integrating the equation of motion and the equation of conservation of water substance. The initial state is taken to be the steady-state solution of the equations in the absence of precipitation. Two models of precipitation development are considered. In the first model, the cloud water is considered to change into rainwater at a prescribed rate, whenever the cloud water content exceeds a given critical value. The raindrops are assumed to have an exponential size distribution and to grow by coalescing with the cloud drops. In the second model, all the cloud water is considered to change, at a given instant, into raindrops of a given fall speed. The main conclusions are: 1) a downdraft develops in the lower levels, 2) the cloud grows vertically consequent upon the development of precipitation, and 3) the updraft and water content execute coupled oscillations.

## Abstract

The effect of precipitation on the properties of a one-dimensional non-entraining cumulus is studied by numerically integrating the equation of motion and the equation of conservation of water substance. The initial state is taken to be the steady-state solution of the equations in the absence of precipitation. Two models of precipitation development are considered. In the first model, the cloud water is considered to change into rainwater at a prescribed rate, whenever the cloud water content exceeds a given critical value. The raindrops are assumed to have an exponential size distribution and to grow by coalescing with the cloud drops. In the second model, all the cloud water is considered to change, at a given instant, into raindrops of a given fall speed. The main conclusions are: 1) a downdraft develops in the lower levels, 2) the cloud grows vertically consequent upon the development of precipitation, and 3) the updraft and water content execute coupled oscillations.

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## Abstract

Calculations show that for spherical hail, a negative value of the range derivative of the logarithm of the ratio of the powers received at 10 and 3 cm wavelengths (*P*
_{10} and *P*
_{3}, respectively) is indicative of the presence of large hail (diameter 1 cm). The negative derivative can occur at the far edge of a boundary between a region of large hail and rain, as shown by Eccles and Atlas (1973). Calculations with a simple model of a hail region, having monodisperse spherical hail at any range, show that a negative derivative of log (*P*
_{10}/*P*
_{3}) can occur embedded in a region of hail if the diameter of the hail and/or the thickness of water coat on it is a function of the range. The results of the calculations are supported by vertical incidence dual-wavelength Doppler radar observations of a hailstorm. Negative values of the range derivative of log (*P*
_{10}/*P*
_{3}) are inferred from the Doppler observations, to occur within a region of hail in which the mean hail diameter was a function of the range. Accordingly, a negative range derivative of log (*P*
_{10}/*P*
_{3}), although indicative of the presence of large hail, is not useful for delimiting the boundaries of a hail region.

## Abstract

Calculations show that for spherical hail, a negative value of the range derivative of the logarithm of the ratio of the powers received at 10 and 3 cm wavelengths (*P*
_{10} and *P*
_{3}, respectively) is indicative of the presence of large hail (diameter 1 cm). The negative derivative can occur at the far edge of a boundary between a region of large hail and rain, as shown by Eccles and Atlas (1973). Calculations with a simple model of a hail region, having monodisperse spherical hail at any range, show that a negative derivative of log (*P*
_{10}/*P*
_{3}) can occur embedded in a region of hail if the diameter of the hail and/or the thickness of water coat on it is a function of the range. The results of the calculations are supported by vertical incidence dual-wavelength Doppler radar observations of a hailstorm. Negative values of the range derivative of log (*P*
_{10}/*P*
_{3}) are inferred from the Doppler observations, to occur within a region of hail in which the mean hail diameter was a function of the range. Accordingly, a negative range derivative of log (*P*
_{10}/*P*
_{3}), although indicative of the presence of large hail, is not useful for delimiting the boundaries of a hail region.

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## Abstract

Data obtained in a thunderstorm with a vertically pointing Doppler radar are analyzed to find the size distribution of raindrops at heights below the 0C level. Drop-size distributions were computed using the updraft *U _{R}
* deduced by Rogers' method and two other updrafts, namely,

*U*− 1 and

_{R}*U*+ 1 m sec

_{R}^{−1}. It is found that the drop-size data at all the heights may be well represented by the exponential equation,

*N*(

*D*) =

*N*

_{0}exp(− Λ

*D*), in which

*N*(

*D*)Δ

*D*is the concentration of drops in the diameter range

*D*to

*D*+ Δ

*D*,

*N*

_{0}= 0.07

*R*

^{0.37}[cm

^{−4}], and Λ = 38

*R*

^{−0.14}[cm

^{−1}],

*R*being the rainfall rate (mm hr

^{−1}). For

*R*≳ 3 mm hr

^{−1}, the distribution is steeper and

*N*

_{0}is greater as compared to the Marshall-Palmer distribution. For radar reflectivity factors

*Z*in the range 1–10

^{5}mm

^{6}m

^{−3}, the relationship between the mean Doppler velocity and

*Z*for the distribution agrees with that given by Rogers to within 1 m sec

^{−1}. The following equations have been found between the water content

*M*, median volume diameter

*D*

_{0}, radar reflectivity factor

*Z*, and the rainfall rate

*R*.

*M*= 0.052

*R*

^{0.94}[gm m

^{−3}],

*D*

_{0}= 0.13

*R*

^{0.14}[mm]

*Z*= 300

*R*

^{1.35}[mm

^{6}m

^{−3}].

## Abstract

Data obtained in a thunderstorm with a vertically pointing Doppler radar are analyzed to find the size distribution of raindrops at heights below the 0C level. Drop-size distributions were computed using the updraft *U _{R}
* deduced by Rogers' method and two other updrafts, namely,

*U*− 1 and

_{R}*U*+ 1 m sec

_{R}^{−1}. It is found that the drop-size data at all the heights may be well represented by the exponential equation,

*N*(

*D*) =

*N*

_{0}exp(− Λ

*D*), in which

*N*(

*D*)Δ

*D*is the concentration of drops in the diameter range

*D*to

*D*+ Δ

*D*,

*N*

_{0}= 0.07

*R*

^{0.37}[cm

^{−4}], and Λ = 38

*R*

^{−0.14}[cm

^{−1}],

*R*being the rainfall rate (mm hr

^{−1}). For

*R*≳ 3 mm hr

^{−1}, the distribution is steeper and

*N*

_{0}is greater as compared to the Marshall-Palmer distribution. For radar reflectivity factors

*Z*in the range 1–10

^{5}mm

^{6}m

^{−3}, the relationship between the mean Doppler velocity and

*Z*for the distribution agrees with that given by Rogers to within 1 m sec

^{−1}. The following equations have been found between the water content

*M*, median volume diameter

*D*

_{0}, radar reflectivity factor

*Z*, and the rainfall rate

*R*.

*M*= 0.052

*R*

^{0.94}[gm m

^{−3}],

*D*

_{0}= 0.13

*R*

^{0.14}[mm]

*Z*= 300

*R*

^{1.35}[mm

^{6}m

^{−3}].

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## Abstract

Some general properties of exponential particle-size distributions are discussed. Especially significant are the corrections to calculated liquid water content, precipitation rate, radar reflectivity, and median volume diameter resulting from truncations of the exponential size spectrum at maximum diameters less than infinity. These properties are used to analyze available snow particle-size distribution data. It is concluded that the equation, *N _{D}
* =

*N*

_{0}exp (−Λ

*D*), describes the size distribution data for snow adequately. In the equation

*N*Δ

_{D}*D*is the concentration of snow particles of melted diameter

*D*to

*D*+ Λ

*D*,

*N*

_{0}= 2.50 × 10

^{3}

*R*

^{−0.94}[mm

^{−1}m

^{−3}], and Λ = 22.9

*R*

^{−0.45}[cm

^{−1}], where

*R*is the precipitation rate in mm hr

^{−1}. These expressions for

*N*

_{0}and Λ differ from those given by Gunn and Marshall, and have the advantage that analytical manipulation of the size distribution equation yields consistent results. The following equations have been found for the precipitation content

*W*, the median volume diameter

*D*

_{0}, and the radar reflectivity factor

*Z*:

*W*= 0.250

*R*

^{0.86}[gm m

^{−3}],

*D*

_{0}= 0.14

*R*

^{0.45}[cm] and

*Z*= 1780

*R*

^{2.21}[mm

^{6}m

^{−3}]. The large exponent in the

*Z*-

*R*equation is related to the fact that the “effective” size of the snow particle-size distribution increases rather rapidly with the precipitation rate.

## Abstract

Some general properties of exponential particle-size distributions are discussed. Especially significant are the corrections to calculated liquid water content, precipitation rate, radar reflectivity, and median volume diameter resulting from truncations of the exponential size spectrum at maximum diameters less than infinity. These properties are used to analyze available snow particle-size distribution data. It is concluded that the equation, *N _{D}
* =

*N*

_{0}exp (−Λ

*D*), describes the size distribution data for snow adequately. In the equation

*N*Δ

_{D}*D*is the concentration of snow particles of melted diameter

*D*to

*D*+ Λ

*D*,

*N*

_{0}= 2.50 × 10

^{3}

*R*

^{−0.94}[mm

^{−1}m

^{−3}], and Λ = 22.9

*R*

^{−0.45}[cm

^{−1}], where

*R*is the precipitation rate in mm hr

^{−1}. These expressions for

*N*

_{0}and Λ differ from those given by Gunn and Marshall, and have the advantage that analytical manipulation of the size distribution equation yields consistent results. The following equations have been found for the precipitation content

*W*, the median volume diameter

*D*

_{0}, and the radar reflectivity factor

*Z*:

*W*= 0.250

*R*

^{0.86}[gm m

^{−3}],

*D*

_{0}= 0.14

*R*

^{0.45}[cm] and

*Z*= 1780

*R*

^{2.21}[mm

^{6}m

^{−3}]. The large exponent in the

*Z*-

*R*equation is related to the fact that the “effective” size of the snow particle-size distribution increases rather rapidly with the precipitation rate.

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## Abstract

The kinetic equation for the evolution of particle size spectra by condensation and coalescence is considered. The condensation rate, the rate of increase of the particle mass ẋ, is taken as (i) *a(t)x* and (ii) *a(t)*, where *a(t)* is an arbitrary non-negative function of the time. In case (i) it is shown that, for a homogeneous kernel, the solution of the kinetic equation for condensation and coalescence can be reduced to that for pure coalescence by simple transformations. In case (ii) the solution is expressed as an infinite series, the terms of which involve convolutions of arbitrary order of the initial distribution and a function of the condensation rate. The central limit theorem of probability theory is used to obtain an expansion for the convolutions, and an approximate analytical expression for the sum of the infinite series is obtained for large *x*. A few numerical evaluations of the solutions are presented.

## Abstract

The kinetic equation for the evolution of particle size spectra by condensation and coalescence is considered. The condensation rate, the rate of increase of the particle mass ẋ, is taken as (i) *a(t)x* and (ii) *a(t)*, where *a(t)* is an arbitrary non-negative function of the time. In case (i) it is shown that, for a homogeneous kernel, the solution of the kinetic equation for condensation and coalescence can be reduced to that for pure coalescence by simple transformations. In case (ii) the solution is expressed as an infinite series, the terms of which involve convolutions of arbitrary order of the initial distribution and a function of the condensation rate. The central limit theorem of probability theory is used to obtain an expansion for the convolutions, and an approximate analytical expression for the sum of the infinite series is obtained for large *x*. A few numerical evaluations of the solutions are presented.

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## Abstract

Equations relating the mean of the Doppler spectrum and the distribution of point velocities, and their spectra are derived under the assumptions that: 1) the scatterers follow the air motion faithfully, 2) the reflectivity is constant, and 3) the beam illumination function is separable. It is found that the three-dimensional spectral density function is strongly attenuated at scales small compared to the beam dimensions, and essentially unaffected at scales large compared to the beam dimensions. Relationships between the one-dimensional longitudinal and transverse spectra of the mean velocity and the three-dimensional spectrum of the point velocities are derived. Numerical computations with a model Kolmogorov-Obukhov turbulence spectrum are performed to illustrate the effects of filtering. Energy at scales small compared to the beam dimensions is attenuated. Energy at scales large compared to the beam dimensions is also reduced, in the case of the one-dimensional spectrum, because small scales in the orthogonal directions contributing to the energy are attenuated by the filtering. The energy depleted from the spectrum of the mean velocity appears as an increased variance of the Doppler spectrum. The ratio of the total energy under the measured spectrum to that under the spectrum of the point velocities is computed as a function of beam dimensions. An equivalent rectangular filter approximation is proposed for computing the one-dimensional spectra. Analytical results are obtained for the longitudinal spectrum and are shown to be in excellent agreement with the numerical results for the actual filter. The use of a spherical volume equal to that of the actual radar pulse volume is shown to be invalid.

## Abstract

Equations relating the mean of the Doppler spectrum and the distribution of point velocities, and their spectra are derived under the assumptions that: 1) the scatterers follow the air motion faithfully, 2) the reflectivity is constant, and 3) the beam illumination function is separable. It is found that the three-dimensional spectral density function is strongly attenuated at scales small compared to the beam dimensions, and essentially unaffected at scales large compared to the beam dimensions. Relationships between the one-dimensional longitudinal and transverse spectra of the mean velocity and the three-dimensional spectrum of the point velocities are derived. Numerical computations with a model Kolmogorov-Obukhov turbulence spectrum are performed to illustrate the effects of filtering. Energy at scales small compared to the beam dimensions is attenuated. Energy at scales large compared to the beam dimensions is also reduced, in the case of the one-dimensional spectrum, because small scales in the orthogonal directions contributing to the energy are attenuated by the filtering. The energy depleted from the spectrum of the mean velocity appears as an increased variance of the Doppler spectrum. The ratio of the total energy under the measured spectrum to that under the spectrum of the point velocities is computed as a function of beam dimensions. An equivalent rectangular filter approximation is proposed for computing the one-dimensional spectra. Analytical results are obtained for the longitudinal spectrum and are shown to be in excellent agreement with the numerical results for the actual filter. The use of a spherical volume equal to that of the actual radar pulse volume is shown to be invalid.

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## Abstract

No abstract available.

## Abstract

No abstract available.

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## Abstract

Analytical solutions for the growth and vertical and horizontal motion of a precipitation particle growing by coalescence with cloud drops are derived under simplified steady-state assumptions. An equation is also developed for the concentration density of a continuous distribution of growing particles.

Assuming that the cloud water content varies linearly with height, and that the fall speed of a drop is proportional to the square root of its diameter, it is shown that the combination of a linearly increasing updraft surmounted by a sharply decreasing one sets a sharp upper limit to the particle size, and sorts the particles horizontally. Particles which spend their entire life in regions of horizontal convergence associated with increasing updraft are packed into a narrower shaft than that in which they originated. Initially smaller particles are carried above into the region of horizontal divergence associated with decreasing updraft and are displaced far to the sides of the cloud core. It is found that when the updraft increases sharply there is a very small range of initial sizes which can grow to fall-out size. These facts are used to suggest that a steady “balance level” (equal reflectivity in rising and falling particles) may be maintained at a height near and below an updraft maximum. Particle size spectra computed from the concentration density equation are continuous and well-behaved for rising, floating and falling particles alike, without necessarily even maximizing for the floating size.

## Abstract

Analytical solutions for the growth and vertical and horizontal motion of a precipitation particle growing by coalescence with cloud drops are derived under simplified steady-state assumptions. An equation is also developed for the concentration density of a continuous distribution of growing particles.

Assuming that the cloud water content varies linearly with height, and that the fall speed of a drop is proportional to the square root of its diameter, it is shown that the combination of a linearly increasing updraft surmounted by a sharply decreasing one sets a sharp upper limit to the particle size, and sorts the particles horizontally. Particles which spend their entire life in regions of horizontal convergence associated with increasing updraft are packed into a narrower shaft than that in which they originated. Initially smaller particles are carried above into the region of horizontal divergence associated with decreasing updraft and are displaced far to the sides of the cloud core. It is found that when the updraft increases sharply there is a very small range of initial sizes which can grow to fall-out size. These facts are used to suggest that a steady “balance level” (equal reflectivity in rising and falling particles) may be maintained at a height near and below an updraft maximum. Particle size spectra computed from the concentration density equation are continuous and well-behaved for rising, floating and falling particles alike, without necessarily even maximizing for the floating size.