Rainfall E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Geophysical Monograph Series
- Sprache: Englisch
Published by the American Geophysical Union as part of the Geophysical Monograph Series, Volume 191.
Rainfall: State of the Science offers the most up-to-date knowledge on the fundamental and practical aspects of rainfall. Each chapter, self-contained and written by prominent scientists in their respective fields, provides three forms of information: fundamental principles, detailed overview of current knowledge and description of existing methods, and emerging techniques and future research directions.
The book discusses
- Rainfall microphysics: raindrop morphodynamics, interactions, size distribution, and evolution
- Rainfall measurement and estimation: ground-based direct measurement (disdrometer and rain gauge), weather radar rainfall estimation, polarimetric radar rainfall estimation, and satellite rainfall estimation
- Statistical analyses: intensity-duration-frequency curves, frequency analysis of extreme events, spatial analyses, simulation and disaggregation, ensemble approach for radar rainfall uncertainty, and uncertainty analysis of satellite rainfall products
The book is tailored to be an indispensable reference for researchers, practitioners, and graduate students who study any aspect of rainfall or utilize rainfall information in various science and engineering disciplines.
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Seitenzahl: 917
Veröffentlichungsjahr: 2013
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CONTENTS
PREFACE
Microphysics, Measurement, and Analyses of Rainfall
1. MICROPHYSICS
2. MEASUREMENT AND ESTIMATION
3. STATISTICAL ANALYSES
Section I: Rainfall Microphysics
Raindrop Morphodynamics
1. INTRODUCTION
2. BACKGROUND ON RAINDROP FLUID DYNAMICS
3. DROP SHAPE
4. DROP OSCILLATION
5. TERMINAL VELOCITY
6. ELECTROSTATIC EFFECTS
7. EXPERIMENTAL TECHNIQUES
The Evolution of Raindrop Spectra: A Review of Microphysical Essentials
1. INTRODUCTION
2. BASICS
3. PHYSICAL DESCRIPTION OF COLLISIONAL INTERACTION OF DROPS
4. COAGULATION
5. A NUMERICAL CASE STUDY
6. BREAKUP
7. COLLISION KERNELS
Raindrop Size Distribution and Evolution
1. INTRODUCTION
2. REPRESENTATION OF COLLISION-INDUCED BREAKUP IN MODELS
3. MEASUREMENTS OF RSDS
4. SUMMARY AND FUTURE WORK
Section II: Rainfall Measurement and Estimation
Ground-Based Direct Measurement
1. INTRODUCTION
2. RAIN GAUGES
3. DISDROMETERS
4. CONCLUDING REMARKS
Radar and Multisensor Rainfall Estimation for Hydrologic Applications
1. INTRODUCTION
2. FOUNDATIONS AND MAJOR SOURCES OF ERROR IN RADAR RAINFALL ESTIMATION
3. REAL-WORLD APPLICATIONS OF RADAR RAINFALL ESTIMATION
4. ADVANCES AND CHALLENGES IN RADAR RAINFALL ESTIMATION
5. MULTISENSOR RAINFALL ESTIMATION
6. CHALLENGES IN MULTISENSOR RAINFALL ESTIMATION
7. CONCLUSIONS
Dual-Polarization Radar Rainfall Estimation
1. INTRODUCTION
2. RAINFALL ESTIMATION USING DUAL-POLARIZATION RADAR MEASUREMENTS
3. GAPS IN CURRENT SCIENCE AND RECENT DEVELOPMENTS
4. SUMMARY
Quantitative Precipitation Estimation From Earth Observation Satellites
1. BACKGROUND
2. DERIVATION OF QUANTITATIVE PRECIPITATION RETRIEVAL METHODOLOGIES
3. APPLICATIONS
4. PRECIPITATION PRODUCT COMPARISONS AND VALIDATION
5. CONCLUSION
Section III: Statistical Analysis
Intensity-Duration-Frequency Curves
1. INTRODUCTION
2. HISTORICAL DEVELOPMENT
3. SPATIAL PRECIPITATION FREQUENCY
4. IDF CURVES FROM OFFICIAL NWS PUBLICATIONS
5. EQUATIONS FOR IDF CURVES
6. DEVELOPMENT OF IDF CURVES FROM RAW DATA
7. PRECIPITATION FREQUENCY DATA SERVER
8. EMERGING ISSUES AND TECHNOLOGIES
Frequency Analysis of Extreme Rainfall Events
1. INTRODUCTION
2. DATA REQUIREMENTS FOR HFA OF RAINFALL DATA
3. STATISTICAL DISTRIBUTIONS USED IN RAINFALL FREQUENCY ANALYSIS AND THEIR PROPERTIES
4. PARAMETER ESTIMATING METHODS
5. PROBABILITY PLOTS, GOODNESS-OF-FIT TESTS, AND CRITERIA
6. NONSTATIONARY RAINFALL FREQUENCY ANALYSIS
7. CASE STUDY: ANNUAL MAXIMUM PRECIPITATIONS IN THE RANDSBURG STATION (CALIFORNIA) AND THE SOUTHERN OSCILLATION INDEX (SOI)
8. BIVARIATE FREQUENCY ANALYSIS OF RAINFALL EVENTS
9. REGIONAL FREQUENCY ANALYSIS OF EXTREME RAINFALL EVENTS
Methods and Data Sources for Spatial Prediction of Rainfall
1. INTRODUCTION
2. DATA SOURCES
3. CASE STUDY: BILOGORA
4. CASE STUDY: ITALY
5. DISCUSSION AND CONCLUSIONS
Rainfall Generation
1. INTRODUCTION
2. STOCHASTIC RAINFALL GENERATION
3. STOCHASTIC RAINFALL GENERATION FOR CLIMATE CHANGE CONDITIONS
4. CONCLUDING REMARKS
Radar-Rainfall Error Models and Ensemble Generators
1. INTRODUCTION
2. SOURCE-SPECIFIC ERROR MODELS
3. MODELS FOR TOTAL ERROR
4. ENSEMBLE GENERATION
5. UNCERTAINTY PROPAGATION
6. OPEN QUESTIONS
7. CLOSING REMARKS
Framework for Satellite Rainfall Product Evaluation
1. INTRODUCTION
2. ART OF EVALUATION
3. EVALUATION METRICS
4. ERROR MODELS
5. HYDROLOGICAL EVALUATION
6. SPECIAL ISSUES: WHICH SATELLITE RAINFALL PRODUCTS ARE BETTER?
7. CONCLUSIONS
AGU Category Index
Index
Geophysical Monograph Series
Published under the aegis of the AGU Books Board
Kenneth R. Minschwaner, Chair; Gray E. Bebout, Joseph E. Borovsky, Kenneth H. Brink, Ralf R. Haese, Robert B. Jackson, W. Berry Lyons, Thomas Nicholson, Andrew Nyblade, Nancy N. Rabalais, A. Surjalal Sharma, and Darrell Strobel, members.
Library of Congress Cataloging-in-Publication Data
Rainfall : state of the science / Firat Y. Testik and Mekonnen Gebremichael, editors.
p. cm. — (Geophysical monograph, ISSN 0065-8448 ; 191)
Includes bibliographical references and index.
ISBN 978-0-87590-481-8 (alk. paper)
1. Rain and rainfall. 2. Rainfall probabilities. 3. Rain and rainfall—Measurement. I. Testik, Firat Y., 1977- II. Gebremichael, Mekonnen.
QC925.R24 2010
551.57′7—dc22
2010049230
ISBN: 978-0-87590-481-8
ISSN: 0065-8448
Cover Image: Raindrops on a window.
Copyright 2010 by the American Geophysical Union
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PREFACE
Rainfall, liquid precipitation, is a critical component of water and energy cycles. It is a critical source of fresh water, sustaining life on Earth, and an important process for energy exchanges between the atmosphere, ocean, and land, determining Earth’s climate. It is central to water supply, agriculture, natural ecosystems, hydroelectric power, industry, drought, flood, and disease hazards for example. Therefore, rainfall is at the heart of social, economical, and political challenges in today’s world. It is a high priority to use advancements in scientific knowledge of rainfall to develop solutions to the water-related challenges faced by society. The three main aspects of rainfall, “rainfall microphysics,” “rainfall measurement and estimation,” and “rainfall statistical analyses,” have been widely studied as individual topics over the years. It is the goal of this book to synthesize all of these aspects to provide an integral picture of the state of the science of rainfall.
This book presents the state of the science of rainfall focusing on three areas: (1) rainfall microphysics, (2) rainfall measurement and estimation, and (3) rainfall statistical analyses. Each part consists of a number of self-contained chapters providing three forms of information: fundamental principles, detailed overview of current knowledge and description of existing methods, and emerging techniques and future research directions. Each book chapter is authored by preeminent researchers in their respective fields and has been reviewed by two renowned researchers within the same field for the scientific accuracy, quality, and completeness of the final content. The book is tailored to be an indispensable reference for researchers, practitioners, and graduate students who study any aspect of rainfall or utilize rainfall information in various science and engineering disciplines.
As editors of this book, we would like to express our utmost gratitude to everyone who has contributed in this publication. We are thankful to all the chapter authors for contributing their expertise and time. We are thankful also to all the reviewers who have selflessly served for the success of this project.
Firat Y. TestikClemson University
Mekonnen GebremichaelUniversity of Connecticut
Microphysics, Measurement, and Analyses of Rainfall
Mekonnen Gebremichael
Department of Civil and Environmental Engineering, University of Connecticut, Storrs, Connecticut, USA
Firat Y. Testik
Department of Civil Engineering, Clemson University, Clemson, South Carolina, USA
Rainfall, liquid precipitation, is a critical component of water and energy cycles. It is a critical source of water for water supply, agriculture, natural ecosystems, hydroelectric power, and industry and is central to issues of drought, flood, and disease hazards. The most desired characteristic of rainfall is the rainfall rate at Earth’s surface. This book, “Rainfall: State of the Science,” aims to synthesize the three main aspects (microphysics, measurement and estimation, and statistical analyses) of rainfall rate estimation efforts to provide an integral picture of this endeavor. In this introductory chapter, we present the issues that will be discussed in detail in the subsequent chapters.
1. MICROPHYSICS
Understanding the microphysics of rainfall is important to accurately estimate rainfall rate from microwave remote sensing and to model the rainfall process in process-based models. Rainfall microphysics, deals with the dynamical processes for individual and populated raindrops throughout their journey from cloud to surface. Rainfall and cloud microstructure is a broad topic, and there are several comprehensive books [Pruppacher and Klett, 1978; Rogers and Yau, 1989; Mason, 1971] devoted to this subject. For the purpose of rainfall rate retrievals, accurate information on the raindrop shape, fall velocity, and raindrop size distribution (DSD) are of particular interest. Therefore, main considerations on rainfall microphysics discussions in this book will be centered on these quantities with a perspective from rainfall rate retrievals. Aside from rainfall rate measurements, these quantities have important applications such as soil erosion studies [Fox, 2004; Fornis et al., 2005], air pollution studies [Mircea et al., 2000], and telecommunications [Panagopoulos and Kanellopoulos, 2002].
Raindrops demonstrate a variety of complex shapes and shape-altering oscillations under the action of a range of surface and body forces. Shapes and fall velocities of raindrops are closely coupled resulting in a dynamic interplay until equilibrium is reached. Raindrop shapes and fall velocities are important input parameters for extracting rainfall information via remote sensing. Polarimetric weather radars utilize vertical to horizontal chord ratios of “equilibrium” raindrop shapes and corresponding “terminal” fall velocities. Consequently, there have been a number of studies on raindrop morphodynamics (i.e., static and dynamic processes related to raindrop shape) over the years [Laws, 1941; Spilhaus, 1948; Gunn, 1949; Gunn and Kinzer, 1949; Savic, 1953; McDonald, 1954; Pruppacher and Pitter, 1971; Green, 1975; Wang and Pruppacher, 1977; Beard, 1977; Beard and Chuang, 1987; Beard et al., 1989]. Jones et al. [this volume] review raindrop morphodynamics, providing a synthesis of information on raindrop shape and related physical processes, including forces shaping the raindrops, raindrop oscillations, and fall velocities.
Raindrops are formed within clouds through collisional interactions of cloud droplets. After formation, raindrops interact with each other via collisions throughout their journey from cloud to surface. These collisions may result in coalescence, breakup, and bounce of colliding drops. As a result of these collision outcomes, DSD continuously evolves with height. Accurate information on DSD at different heights is important for obtaining accurate rainfall information via remote sensing. Numerical simulations based on the stochastic coalescence/breakup equation and laboratory models of governing processes [Low and List, 1982a, 1982b] are used to obtain information on the DSD evolution with height [e.g., Gillespie and List, 1978; List and McFarquhar, 1990; McFarquhar, 2004]. Beheng [this volume] discusses the formation of raindrops from cloud droplets and collisional interactions of raindrops that result in an evolution of the raindrop size distribution. Environmental interactions are omitted.
Following the landmark study by Marshall and Palmer [1948], various distributions have been used to represent DSD, including exponential, Gaussian, and lognormal distributions. The assumed form of DSD plays a critical role in rainfall rate retrieval from both ground- and space-based systems. Various numerical simulations of DSD evolution under the action of collisional interactions have shown evidence for an equilibrium form of DSD after sufficient evolution time is given [e.g., Valdez and Young, 1985; List et al., 1987; List and McFarquhar, 1990; Hu and Srivastava, 1995]. However, field observations have not verified the occurrence of an equilibrium DSD as predicted by these numerical simulations. McFarquhar [this volume] provides an overview of observed raindrop size distributions at both the ground and aloft as well as the evolution of the raindrop size distributions throughout the rain shaft from numerical simulations.
2. MEASUREMENT AND ESTIMATION
Information on rainfall properties can be obtained by means of different observing systems and associated algorithms: direct in situ sensors (i.e., rain gauges and disdrometers), ground-based remote sensors (i.e., weather radars), and space-based remote sensors (i.e., radars and infrared sensors). Each sensor has its own strengths and limitations.
Rain gauges and disdrometers provide direct in situ point measurements of rainfall properties at high temporal resolution. Rain gauge measurements of rainfall rate continue to be the main basis for “calibrating” rainfall remote sensing algorithms and for numerous research and operational applications that require rainfall data. However, even at their point measurement scales, rain gauge rainfall measurements are subject to systematic and random measurement errors, the most important of which are the following: the drift of rainfall particles due to wind field deformation around the gauge, losses caused by wetting of the inner walls of the gauge, evaporation of water accumulated in the gauge container, splashing of raindrops out or into the gauge, calibration-related errors, malfunctioning problems, poor location selection, and local random errors. Disdrometers measure DSD that describes the rainfall microstructure. The DSD data have been widely used in studying soil erosion and rainfall microphysical properties and, perhaps most importantly, derivation of radar rainfall retrieval algorithms. The DSD measurements can be affected by various sources of errors, which can be grouped into instrumental, sampling, and observational errors. Habib et al. [this volume] present an overview of the different types of rain gauges and disdrometers, discuss the major sources of uncertainties that contaminate measurements at the local point scale, and describe the recently developed methods for automatic quality control of the rain gauge data.
The availability of rainfall estimates from conventional ground-based scanning weather radars (i.e., single-polarization radar systems), such as the U.S. Weather Surveillance Radars-1988 Doppler (WSR-88D) radars, at high space-time resolutions and over large areas has greatly advanced our quantitative information on the space-time variability of rainfall. However, because radar measures volumetric reflectivity of hydrometeors aloft rather than rainfall near the ground, radar rainfall estimation is inherently subject to various sources of error. The major sources and possible practical consequences of these errors have been well recognized and discussed by many researchers [e.g., Wilson and Brandes, 1979; Zawadzki, 1982; Austin, 1987; Krajewski and Smith, 2002; Jordan et al., 2003; Krajewski et al., 2010]. Recent work on improving the accuracy of rainfall estimates from conventional weather radars consists of incorporating rain gauge measurements to remove the bias in the radar rainfall estimates and using multiple radar rainfall fields whenever possible. Seo et al. [this volume] describe the foundations of radar and multisensor rainfall estimation, recent advances and notable applications, and outstanding issues and areas of research that must be addressed to meet the needs of forecasting in various applications such as hydrology.
As part of the modernization of the WSR-88D, the U.S. National Weather Service and other agencies have decided to add a polarimetric capability to existing conventional single-polarization radars. Dual polarization provides additional information compared to single-polarization radar systems, which helps to significantly improve the accuracy of radar rainfall estimates. The three polarimetric variables that are often used in rainfall estimation are the radar reflectivity at horizontal polarization, the differential reflectivity (defined as the difference between reflectivities at horizontal and vertical polarizations), and the differential phase (defined as the difference between the phases of the radar signals at orthogonal polarizations). An overview of the methods of rainfall estimation from these variables is presented by Cifelli and Chandrasekar [this volume].
Remote sensing from a space platform provides a unique opportunity to obtain spatial fields of rainfall information over large areas of Earth. During the last two decades, satellite-based instruments have been designed to collect observations mainly at thermal infrared (IR) and microwave (MW) wavelengths that can be used to estimate rainfall rates. Observations in the IR band are available in passive modes from (near) polar orbiting (revisit times of 1–2 days) and geostationary orbits (revisit times of 15–30 min), while observations in the passive and active MW band are only available from the (near) polar-orbiting satellites. A number of algorithms have been developed to estimate rainfall rates by combining information from the more accurate (but less frequent) MW observations with the more frequent (but less accurate) IR observations to take advantage of the complementary strengths [Sorooshian et al., 2000; Scofield and Kuligowski, 2003; Joyce et al., 2004; Turk and Miller, 2005; Huffman et al., 2007; Ushio and Kachi, 2010]. Satellite rainfall estimates are subject to a variety of error sources (gaps in revisiting times, poor direct relationship between MW cloud top measurements and rainfall rate, atmospheric effects that modify the radiation field, etc.). The errors increase with increasing space-time resolution and depend largely on the algorithm technique, type, and number of satellite sensors used and the study region [e.g., Hong et al., 2004; Gottschalck et al., 2005; Brown, 2006; Ebert et al., 2007; Tian et al., 2007; Bitew and Gebremichael, 2010; Dinku et al., 2010; Sapiano et al., 2010]. Ongoing research and development continues to address the accuracy and the resolution (temporal and spatial) of these estimates. Kidd et al. [this volume] cover the basis of the satellite systems used in the observation of rainfall and the processing of these measurements to generate rainfall estimates, discuss research challenges, and provide research and development recommendations.
3. STATISTICAL ANALYSES
Various statistical techniques are often applied to rainfall data depending on the application and source of data. Commonly employed statistical analyses are extreme event analysis, spatial interpolation of point rainfall, rainfall generation, and uncertainty analysis of remote sensing rainfall estimates.
Statistical analysis of extreme rainfall events is useful for a number of engineering applications including hydraulic design (culverts and storm sewers) and landslide hazard evaluations. Intensity-duration-frequency (IDF) curves, with areal reduction factors, are commonly used for design storm calculations. For any prescribed rainfall duration (which depends on the time of concentration for the watershed) and return period (which is often set as a standard value depending on the purpose and failure consequence of the hydraulic structure), the corresponding design rainfall intensity is obtained from regional IDF curves. The common method of developing IDF consists of the following steps: getting the annual maximum series of rainfall intensity for a given duration, using distributions (parametric or nonparametric) to find rainfall intensity for different return periods, and repeating the above steps for different durations. This method has recently come under criticism as it conveniently ignores the joint probability distribution among the rainfall characteristics: depth, intensity, and duration. The difficulty in modeling the joint distributions is that most parametric multivariate distributions are unable to handle rainfall because of the heavy-tailed distributions in the rainfall characteristics. The recently emerging technique of copula [Sklar, 1959] has shown promise in overcoming this difficulty because of its ability to model the dependence structure independently of the marginal distributions. Recent studies have successfully used the copula technique to model the joint distribution among rainfall depth, intensity, and duration [De Michele and Salvadori, 2003; Salvadori and De Michele, 2004a, 2004b; Zhang and Singh, 2007; Kao and Govindaraju, 2007, 2008; Wang et al., 2010]. Durrans [this volume] presents an overview of the historical development of IDF, common methods for constructing IDF, and emerging new methods.
The frequency (or return period) analysis of extreme events is important to develop IDF curves and to test for any trends in the extremes. A number of parametric distributions (Gumbel, generalized extreme value, lognormal, log-Pearson type 3, Halphen, and generalized logistics) have been developed over the years to model the extreme rainfall events. There are two main statistical approaches to fit these distributions. The first approach applies to annual maxima of time series. The second approach looks at exceedances over high threshold, also known as the “peaks over threshold” approach. Prior to any statistical analyses, the data need to be checked for any outlier, dependence, and stationarity. Focusing on the first approach, El Adlouni and Ouarda [this volume] present detailed information on data preparation (detection and treatment of outliers, independence, and stationarity), the parametric distributions (and associated parameter estimation techniques) in the case of stationary time series, and modeling of nonstationary time series.
Spatial rainfall analysis is performed to estimate areal rainfall from point rain gauge data, or to estimate rainfall value at a site based on rainfall measured at another site and auxiliary information, or to generate a spatial pattern. Hengl et al. [this volume] present the spatial analysis techniques used in rainfall, with programming codes to help interested users apply the techniques. Stochastic rainfall generation is performed to generate long time series of rainfall data for a variety of applications including probabilistic failure assessment of natural or man-made systems where rain is an important input. Sharma and Mehrotra [this volume] present an overview of stochastic generation of rainfall, with a focus on daily and subdaily rainfall generation at point and multiple locations, for the current climate assuming climatic stationarity, as well as for future climates using exogenous inputs simulated using general circulation models under assumed greenhouse gas emission scenarios.
Remote sensing rainfall estimates are subject to systematic and random errors from various sources, some of which are inherent to the observation system and are unavoidable. Operational remote sensing rainfall products are deterministic and do not contain quantitative information on the level of the estimation errors. This has led to the current situation in which those who use remote sensing rainfall estimates know that there are significant errors in the estimates, but they have no quantitative information about the magnitudes of the estimation errors. Consequently, there are no mechanisms to account for the uncertainty of remote sensing rainfall estimates in applications and decision making. A possible solution to this major problem is to construct an error model that characterizes the conditional distribution of actual rainfall rate for any given remote sensing rainfall estimate. Mandapaka and Germann [this volume] present the advances in the area of weather radar rainfall error modeling that have taken place over the past decade. Compared to weather radar rainfall estimates, the satellite rainfall estimates are subject to additional error sources and therefore have higher estimation errors. Gebremichael [this volume] presents a recommended standard framework for quantifying errors in satellite rainfall estimates, reviews existing error models and presents emerging ones, and performs quantitative assessment of the utility of satellite rainfall estimates for hydrological applications in selected regions.
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M. Gebremichael, Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT 06269-2037, USA.
F. Y. Testik, Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA. ([email protected])
Section I
Rainfall Microphysics
Raindrop Morphodynamics
B. K. Jones
Department of Mechanical Engineering, Columbia University, New York, New York, USA
J. R. Saylor
Department of Mechanical Engineering, Clemson University, Clemson, South Carolina, USA
F. Y. Testik
Department of Civil Engineering, Clemson University, Clemson, South Carolina, USA
In the absence of forces other than surface tension, a water drop will attain a perfectly spherical shape. Raindrops experience a range of forces, including those due to fluid flow (both inside and outside the drop), hydrostatic forces, and electrostatic forces. A falling raindrop deviates in shape from spherical, becoming a flattened oblate spheroid, a shape that becomes more prominent as the raindrop diameter increases. This shape is characterized by a chord ratio, which is the ratio of the height to the width of the raindrop. The drop shape is often variable, oscillating because of excitation of the natural frequencies of the drop by the flow of fluid around the drop and through interactions between the natural frequency of the drop and vortex shedding in the wake of the drop. These interactions make raindrop morphodynamics, the study of the dynamic and stable raindrop shape, an especially rich problem. Drop collisions also affect the transient behavior of drop shape. Polarimetric radar techniques have further motivated studies of raindrop morphodynamics, since knowledge of raindrop shape can be utilized to improve rain rate retrievals using these radars. Experimentation and analytical efforts have explored several facets of raindrop morphodynamical behavior, including raindrop fall speeds, nonoscillating and oscillating shapes, chord ratio versus diameter relationships, oscillation frequencies, and the preferred harmonic modes, for example. Herein, we provide a survey of the current state of knowledge of these aspects of raindrop morphodynamics.
1. INTRODUCTION
As water vapor in the atmosphere condenses, liquid droplets are initially sufficiently small to remain aloft, entrained in air currents. The motion of these cloud droplets causes them to collide with one another and form either permanent unions or smaller fragment droplets. In this manner, some drops increase in mass until the force of gravity exceeds the momentum available from the air motion, and they begin to fall. This collision and fragmentation occurs in falling drops as well. In fact, an upper size limit is determined as some falling drops coalesce until breakup invariably occurs due to hydrodynamic instability [Pruppacher andPitter, 1971] or because of drop-drop collision. If they remain in the liquid phase, these hydrometeors are referred to as raindrops. Due to their fluid nature, raindrops assume a variety of complex shapes and shape-altering oscillations during free fall. The study of this static and dynamic behavior is referred to as raindrop “morphodynamics” and is based primarily in fluid mechanics. The characteristic shape of a nonoscillating (hereafter “equilibrium”) raindrop is shown in Figure 1.
Figure 1. Water drop levitated in a vertical wind tunnel illustrating the characteristic shape of a quiescent raindrop.
Voluminous work involving a variety of theoretical modeling and experimentation can be found on this subject in the literature (see Testik and Barros [2007] for an extensive review). Experimenters have utilized high-speed photography of natural rain [Testik et al., 2006], for example, or devised creative methods to elucidate real raindrop behavior from water drops floating in wind tunnels [Beard and Pruppacher, 1969; Pruppacher and Beard, 1970; Kamra et al., 1986; Saylor and Jones, 2005; Szakall et al., 2009; Jones and Saylor, 2009] or falling from high stairwells, towers, or highway bridges [Andsager et al., 1999; Thurai and Bringi, 2005]. Theoretical models of raindrop shape reflect the solution of complex differential equations that rely on prior empirical observations for boundary conditions and validation. The accuracy of these calculated shapes, when compared with observations of real raindrops, has chronologically increased as researchers have improved upon the assumptions, techniques, and errors of their predecessors.
Aside from scientific novelty, the study of raindrop morphodynamics is an important aspect of precipitation science, global hydrology, weather radar science [Chandrasekar et al., 2008], and satellite and terrestrial communication techniques [Thurai and Bringi, 2005]. For example, accurate evaluation of dual-polarization (also “polarimetric”) weather radar relies upon precise knowledge of raindrop shapes. Raindrops can also attenuate and disrupt wireless communication links operating at or above microwave frequencies, so correcting for these errors may be possible with more advanced knowledge of raindrop morphodynamic behavior [Allnutt, 1989].
The remainder of this chapter encompasses six sections. A brief background on the fluid dynamics of raindrops is provided in section 2. The equilibrium raindrop shape is introduced in section 3. Raindrop oscillations and resulting shape changes are discussed in section 4. The effect of raindrop shape on terminal fall velocity ut is presented in section 5, and in section 6, the effect of electrical fields on raindrop shape is discussed. Experimental techniques used to study raindrops are presented in section 7.
2. BACKGROUND ON RAINDROP FLUID DYNAMICS
The airflow past a raindrop and the water flow inside a raindrop in free fall are governed by the continuity equation and the Navier-Stokes equations of motion, subject to the appropriate boundary conditions. However, being a nonlinear system of coupled partial differential equations, analytical solution of the Navier-Stokes equations is currently prohibitively complex unless simplifications can be introduced. Discussion of the Navier-Stokes equations and their theoretical treatment in the context of raindrops is given by Pruppacher and Klett [1997] and is not presented here. Further discussion of flows relevant to raindrops can be found in various graduate-level fluid mechanics textbooks [e.g., Batchelor, 1967; Landau and Lifshitz, 1959].
There are primarily three dimensionless parameters pertinent to the morphodynamics of raindrops in free fall: the Reynolds number Re, Weber number We, and Strouhal number St, defined as
(1)
(2)
(3)
where U is the relative velocity (fall speed) between the airstream and the raindrop during free fall, d is the raindrop diameter, ν and ρa are the kinematic viscosity and density of air, respectively, σ is the surface tension of water, and fw is the frequency of vortex shedding in the drop wake, relevant to larger raindrops (discussed below). Because of the variable nature of raindrop shape, the drop diameter d refers to the diameter of an equivalent volume sphere. The variation in morphodynamic behavior that raindrops exhibit can be correlated to these dimensionless parameters, since the ratios in equations (1)–(3) represent the relative magnitudes of underlying fluid forces. Specifically, Re is the ratio of inertial to viscous forces, while We is the ratio of inertial to surface tension forces. The Strouhal number St is the dimensionless frequency of periodic behavior in the raindrop wake, which arises for d > 1 mm. Based on similarity arguments [see, e.g., Barenblatt, 2003], by matching these parameters in fluid systems other than air/water, inferences can be made regarding raindrop behavior, an approach which can simplify experimentation (see below).
Table 1. Approximate Values of Re and We as a Function of Drop Diameter d for Raindrops
d
(
mm
)
Re
We
0.2
10
8.6(10
–4
)
1.0
400
0.3
2.0
1380
1.4
3.0
2510
3.1
4.0
3670
5.0
5.0
4810
6.8
6.0
5920
8.6
Because a drop of given diameter has a nominally fixed maximum (terminal) fall velocity (see section 5) (ut, U in equations (1) and (2)) is a function of d. Hence, Re and We are essentially determined by d, except after drop collisions when U is readjusting to ut. The values that Re and We attain at ut for a span of d representative of raindrops is presented in Table 1, showing that the range of Re and We for raindrops spans nearly four orders of magnitude, an indication of the widely varying balance of forces.
The characteristics of the airflow around a falling raindrop vary significantly with d and thus fall speed and the governing dimensionless parameters. Laboratory visualizations of freely falling drops suggest that for raindrops with Re 210–270, a separated wake develops in the downstream region of the drop [Margarvey and Bishop, 1961]. The presence of this wake region alters the pressure distribution around the raindrop, inducing static and dynamic changes in the raindrop shape. Direct observation of this coupling between shape and wake behavior is a significant experimental challenge, and analogous fluid systems have been studied to elucidate the nature of the relationship for raindrops. For example, Magarvey and MacLatchy [1965] described the wakes of solid spheres falling through a liquid bulk, a fluid system which deviates from the raindrop case due to the rigidity of the sphere surface; for liquid drops, the deforming surface is a significant dissipator of energy. While these deviations somewhat complicate an exact comparison with raindrops, on the other hand, rigid sphere studies facilitate isolation of the wake formation mechanisms from the effects of the liquid drop free surface.
Figure 2. Temporal evolution (from left to right) of vortex shedding pattern in the wake of a solid sphere at Re < 800, as illustrated by Sakamoto and Haniu [1990]. From Sakamoto and Haniu [1990]. Copyright ASME.
Table 2. Classification of Wakes of Freely Falling Liquid Drops and Spheresa
Figure 3. Calculated raindrop shapes from the numerical model due to Beard and Chuang [1987]. From Beard and Chuang [1987]. Copyright 1987 American Meteorological Society.
3. DROP SHAPE
The shape of a falling raindrop is determined by the mechanical equilibrium of the liquid-gas interface defining its outer surface. During free fall, an aerodynamic pressure difference arises between the upper and lower poles and the equator of the raindrop, in addition to an internal circulation because of the no-slip boundary condition at the drop surface. The resulting forces, together with electrostatic forces, internal hydrostatic pressure and surface tension, balance to produce an equilibrium shape resembling a flattened sphere with a wide horizontal base and a smoothly curved upper surface. This shape varies with d, consequently small drops are essentially spherical while larger drops are more distorted. Figure 3 shows this effect, best characterized by the variation in the raindrop chord axis ratio, defined as the ratio of the vertical extent a to the horizontal extent b of the drop, or
(4)
Because of this flattening of raindrop shape with size, the ratio α decreases with increasing d. This trend persists until fragmentation occurs, typically around d ≈ 6–8 mm [Pruppacher and Pitter, 1971], although extraordinary instances of d larger than 8.8 mm [Hobbs and Rangno, 2004] and 10 mm [Takahashi et al., 1995] have been observed in tropical clouds.
This predictable variation of α with d is the key principle behind polarimetric radar techniques [Seliga and Bringi, 1976]. Measurement of rain rate R using traditional single-polarization radar involves transmitting a microwave signal and measuring the intensity of the echo backscattered by raindrops. This intensity determines a reflectivity factor Z, which is used to estimate parameters of the drop size distribution. If the Rayleigh approximation is made for the backscattering cross-section of raindrops, the reflectivity factor Z and R are related to this drop size distribution by [Doviak and Zrnić, 1984]
(5)
(6)
where N(D) is the drop size distribution (DSD), ut is the terminal velocity, and D is the drop diameter (equivalent to d).
The DSD is typically modeled as the Marshall-Palmer spectrum
(7)
having two parameters Λ and N0. Substituting equation (7) into equation (5) yields, after integration,
(8)
Equation (8) reveals the primary obstacle to using single-polarization radar for the measurement of R: how to determine the two unknown parameters Λ and N0 from the single measurement Z? The solution is to utilize dual-polarization radars. Because of the importance of drop shape to this measurement method, we now describe how dual-polarization radar measurement of rain is implemented.
In the most common implementation of this technique, the transmitted radar signal is repeatedly switched between a horizontal and vertical polarization so that two reflectivity factors, ZH and ZV, are measured by the receiver. The ratio of these factors gives the differential reflectivity ZDR, defined as
(9)
This differential reflectivity varies with the specific drop sizes that are aloft during sensing, since different-sized drops exhibit distinctly different α values as shown in Figure 3. Following the treatment of Ulbrich [1986], R is related to ZDR by equation (9) and the definitions of ZH and ZV, given by
(10)
where σH,V is [Gans, 1912]
(11)
(12)
(13)
Based upon laboratory observations, Pruppacher and Pitter [1971] described the variation with d of raindrop shapes as a continuum with three distinct diameter ranges: Class I (d < 0.25 mm), Class II (0.25 mm ≤ d ≤ 1 mm), and Class III (d > 1 mm). Specifically, Class I drops exhibit no detectable distortion from sphericity. These shapes are dominated by surface tension, which effectively minimizes the energy and surface area of the drop, requiring a spherical shape. Class II drops exhibit a slight distortion with a discernible increase in radius of curvature of the lower hemisphere, a shape termed “oblate spheroidal.” Class III category drops show further, marked distortion with increasing diameter, the oblate spheroid shapes exhibiting an increased flattening of the lower surface corresponding to a reduction in axis ratio. These observations are summarized in Table 3.
Early work on the development of a mathematical relationship between α and d focused on confirming and clarifying the relative roles of five physical factors: (1) surface tension, which forces a more spherical shape; (2) internal hydrostatic pressure, a vertical pressure gradient within the drop, acting outward against surface tension; (3) external aerodynamic pressure, which flattens the raindrop as it creates an increase in air pressure at the base and a decrement elsewhere; (4) internal circulation, which creates a toroidal-vortex flow within the drop, inducing complex effects on shape; and (5) electrostatic forces that may accentuate or suppress oblateness depending ondrop electrical charge and field conditions [Lenard, 1904; McDonald, 1954a]. The dominance of the first three factors has been established; however, the role of internal circulation and electrostatic forces in controlling drop shape has yet to be fully understood [Testik and Barros, 2007]. This is because reported results on the amplitudes of internal circulation in raindrops falling at terminal velocity are rather contradictory [Blanchard, 1949; McDonald, 1954a; Garner and Lane, 1959; Foote, 1969; Pruppacher and Beard, 1970]. Additionally, electrostatic forces may have a strong, nonlinear effect on drop distortion in certain thunderstorm conditions [Beard et al., 1989a; Bhalwankar and Kamra, 2007; Beard et al., 2004].
Table 3. Classification of Drop Distortion With Size (d)
Class
d
(mm)
Drop Shape
I
<0.25
No detectable distortion
II
0.25–1.0
Slightly aspherical
III
>1.0
Markedly oblate spheroidal
Two theoretical approaches have been used in raindrop shape models: “gravity” models and “perturbation” models. Gravity models derive an α versus d relationship from a balance of gravity and surface energy [Beard, 1984a], or surface tension and either external or internal pressure [Green, 1975; Spilhaus, 1948], often attaining considerable accuracy despite their relative simplicity. By comparison, the more rigorous perturbation models [Imai, 1950; Savic, 1953; Pruppacher and Pitter, 1971] utilize Laplace’s pressure balance, which relates the curvature at each point on the drop surface to the internal and external pressures by
(14)
where R1 and R2 give the radii of curvature, and Δp gives the pressure difference across the drop surface (for derivation, see Landau and Lifshitz [1959]). The system of differential equations in equation (14) describes the complete drop silhouette. Typically, solutions to this system of equations are obtained numerically and incorporate empirical pressure measurements from wind tunnel data, a method first proposed by Savic [1953]. Pressure measurements from rigid spheres were used by Pruppacher and Pitter [1971] and others until more recently, when the technique was adapted by Beard and Chuang [1987] to account for an altered pressure field from drop distortions.
Figure 4. Comparison of the Beard and Chuang [1987] raindrop shape model (solid line) with the silhouette of a 6-mm wind tunnel drop (shadow) and the model of Pruppacher and Pitter [1971] (dashed line). From Szakallet al. [2009]. Copyright 2009 American Meteorological Society.
Their model, which compares well with field [Chandrasekar et al., 1988; Bringi et al., 1998] and laboratory [Szakall et al., 2009; Thurai et al., 2007] observations, is widely accepted as the most realistic for determining raindrop equilibrium shapes. A recent validation by Szakall et al. [2009] is shown in Figure 4.
4. DROP OSCILLATION
Schmidt [1913] was the first to observe oscillations in the shape of raindrops, and it is well known that d ≥ 1 mm (Class III) drops may oscillate during free fall so that their shapes vary about the equilibrium shape [Gunn, 1949; Jones, 1959]. This complicates the α versus d relationship because instantaneous α measurements often scatter widely. Hence, the accurate interpretation of radar backscatter from oscillating drops requires precise knowledge of the “time-average” (mean) axis ratio as a function of drop diameter. Considerable research has been conducted to elucidate this behavior. Figure 5 shows a sequence of superposed images of a single oscillating drop, with the instants of maximum and minimum amplitude shown approximately at c and h, and f and j, respectively.
Rayleigh [1879] showed that drop oscillations occur at n discrete harmonics and with frequency f decreasing with d according to
(15)
(16)
Figure 5. Superposed images of a single oscillating water drop slowly rising in the test section of a vertical wind tunnel (d ≈ 2.3 mm, frame rate =109 Hz).
Figure 7. Views of the fundamental spherical harmonic (2, m). Two views of the fundamental transverse mode are shown, illustrating the dependence of α on the orientation ϕ of the drop. From Beard et al. [1989b]. Reprinted by permission from Macmillan Publishers Ltd. Copyright 1989.
Beard and Kubesh [1991] examined the theoretical shapes shown in Figure 6 and identified the specific α variation for each mode. They determined that the transverse (2,1) mode gives a strictly positive variation in α toward unity. For this mode, the drop chords exhibit two variances, depending on the viewing angle ϕ: either together, giving no change in α (see bottom-left sketch in Figure 7), or independently with b remaining static and a varying positively (bottom-right sketch of Figure 7). In contrast, axisymmetric oscillations always produce a two-sided variation because both a and b vary in an opposing manner, independent of viewing angle. The horizontal mode (see top-right sketch in Figure 7) similarly produces a two-sided variation because of static a for this mode. Thus, with respect to the equilibrium α, the mean α of an oscillating drop may or may not shift, depending on the prevailing mode. As such, modal behavior can be inferred from the scatter and mean of α measurements of oscillating drops.
Because the shift can be significant and varying, a precise formulation of mean α versus d for radar and microwave scattering applications is important. Seliga and Bringi [1976] found polarimetric radar signals altered by 30% because of uncertainty in mean α, leading to erroneous estimates of drop size and rainfall rate [Kubesh and Beard, 1993].
Although the pioneering work of Rayleigh
