Area Time Integral Research Articles

Dynamic characteristics of droplet impact on vibrating superhydrophobic substrate

The vibration of solids is ubiquitous in nature and in industrial applications and gives rise to alternative droplet dynamics during impact. Using many-body dissipative particle dynamics, we investigate the impact of droplets on superhydrophobic solid surfaces vibrating in the vertical direction at a vibration period similar to the contact time. Specifically, we study the influence of the impact phase and vibration frequency. We evaluate the influence from the aspects of maximum spreading diameter, the solid–liquid contact time and area, and the momentum variation during the impact. To quantitatively evaluate the solid–liquid contact, we introduce the area-time integral, which is the integral of the contact area over the whole contact time. It is meaningful when the heat exchange between solid and liquid is considered. One characteristic phenomenon of droplets impacting vibrating substrate is that multiple contacts may occur before the final rebound. Unlike previous studies defining the contact time as the time span from the first impact to the final detachment, we define the contact time as the summation of each individual contact time. Using this definition, we show that the discontinuity at the critical impact phase disappears. The fact that the area-time integral also changes continually with the impact phase supports the assumption that the effect of impact phase on the solid–liquid contact may be continuous. Moreover, we show that the probability of impact phase is affected by the vibrating frequency and use it to calculate the weighted averaged outcome when the impact phase is not controlled. This study not only offers insights into the physics of droplet impact on vibrating surfaces but also can be used to guide the design of surfaces to achieve manageable wetting using vibration.

Seasonal characteristics of storms over the Indian subcontinent

Storms are convective cells responsible for the major fraction of convective precipitation. Here, the pre-monsoon and monsoon season characteristics of storms are reported at Lucknow, Patna, Bhopal, and Nagpur in India using equivalent radar reflectivity factor (hbox {Z}_e) given at these radar locations. It is observed that the lifetime, speed of propagation, area, volume, echo top height and thickness lie in ranges 0.3–3 h, 5–60 km hbox {h}^{-1}, 4–184 hbox {km}^2, 8–1600 hbox {km}^3, 2–14 km, and 0.5–16 km respectively. For both seasons, the relationships between radar estimated rain volume (RERV; range 10^4–10^7hbox {m}^3) and area-time integral (ATI; range 1–100 hbox {km}^2 h) are established which are considered as a representative of total precipitation resulted from an individual storm during its life cycle. The results from statistical analysis of RERV-ATI pairs suggest that storms at Lucknow have similar seasonal characteristics at 87% confidence interval while other locations exhibit dissimilarities. In addition, the vertical profiles of radar reflectivity (VPRRs) of storms are constructed at their life phases, namely cumulus, mature and dissipation. It is concluded that the vertical hbox {Z}_e gradient in mixed-phase region (5–8 km) is lower (2–2.9 dBZ hbox {km}^{-1}) at cumulus and dissipation phases than at mature phase (3.6–4.4 dBZ hbox {km}^{-1}) in monsoon season. For pre-monsoon season, this gradient lies between 3.3–5.2 dBZ hbox {km}^{-1} at mature phase. Our results are of great importance for advancing knowledge about storm-scale, which has implications in short-range weather forecasting as well as developing new convective parametrization schemes.

Storm characteristics and precipitation estimates of monsoonal clouds using C-band polarimetric radar over Northwest India

Storm is a convective cell much smaller than a mesoscale convective system (MCS) but typically larger than a cumulonimbus cloud and heavy precipitation and lightning are often associated with it. Storms contribute major fraction of the convective precipitation in MCSs. Storm characteristics and precipitation estimates around New Delhi (28.6°N, 77.2°E; an Indian land location) during June–September period of the year 2013 are reported here using data of C-band polarimetric Doppler weather radar. Storms, defined based on radar reflectivity thresholds (30 dBZ for simple storms and 40 dBZ for intense storms), are tracked and their properties are extracted. Our results show that about 80% of storms exist for 1 h or less. The areas of 90% of simple storms are less than 100 km2 and the largest area averaged over storm lifespan does not exceed 400 km2. The majority of storms (> 80%) move with speeds less than 30 km h−1. About 60–65% of simple/intense storms have echo top heights between 6 and 10 km, while only few of them exceed 17 km. The values of average thickness of simple and intense storms lie between ~ 2–10 and ~ 1–7 km, respectively. It is not the vertical extent of a storm but its area-time integral that correlates better with the total precipitation amount. Around the New Delhi area, daily accumulated precipitation derived from relations incorporating polarimetric variables is in good agreement with the rain gauge measurements while that obtained from relations based on radar reflectivity factor (Zh) alone highly underestimates precipitation. This suggests that polarimetric capability is needed in Doppler weather radars to get the realistic precipitation estimates. The mean precipitation water content derived from Zh (~ 0.96 g m−3) is about 30–40% less compared to that derived from polarimetric relations. Our findings on storm properties have implications for cloud parameterizations and in short-term weather forecasting.

Dimension Characteristics and Precipitation Efficiency of Cumulonimbus Clouds in the Region Far South from the Mei-Yu Front over the Eastern Asian Continent

Abstract Dimension characteristics in precipitation properties of cumulonimbus clouds are basic parameters in understanding the vertical transport of water vapor in the atmosphere. In this study, the dimension characteristics and precipitation efficiency of cumulonimbus clouds observed in the Global Energy and Water Cycle Experiment (GEWEX) Asian Monsoon Experiment (GAME) Huaihe River Basin Experiment (HUBEX) are studied using data from X-band Doppler radars and upper-air soundings. The maximum echo area (EAmax) of the cumulonimbus clouds ranged from 0.5 to 470 km2, and the maximum echo top (ETmax) ranged from 2 to 19 km. The total number of cells (TNC) within the cumulonimbus clouds over their lifetime was from 1 to 25. The ETmax, TNC, area time integral (ATI), and total rainfall amount (Rtot) strongly correlate with the EAmax of the cumulonimbus clouds. The cell-averaged ATI (ATIcell = ATI/TNC), maximum rainfall intensity (RImax), and cell-averaged rainfall amount (Rcell = Rtot/TNC) increase when the EAmax is smaller than 100 km2. On the other hand, they are almost constant when the EAmax is larger than 100 km2. The rain productivity of small clouds (<100 km2 in EAmax) increases not only by the increase of the TNC but also by the intensification of cells, while that of large cumulonimbus clouds (>100 km2 in EAmax) increases by the increase of the TNC rather than by the intensification of cells. In the present study, precipitation efficiency (ɛp) is defined as the ratio of the total rainfall amount (Rtot) to the total water vapor amount ingested into the cloud through the cloud base (Vtot). The ɛp was calculated for six clouds whose vertical velocity data at the cloud-base level were deduced by dual-Doppler analysis throughout their lifetime. The ɛp ranged from 0.03% to 9.31% and exhibited a strong positive correlation with the EAmax. This indicates that more than 90% of the water vapor that enters the clouds through the cloud base is consumed to moisten the atmosphere and less than 10% is converted to precipitation and returned to the ground. The cumulonimbus clouds in the region far south from the mei-yu front over the eastern Asian continent efficiently transport water vertically and humidify the upper troposphere.

The Relation between Rainfall and Area–Time Integrals at the Transition from an Arid to an Equatorial Climate

Abstract The area–time integral (ATI) method has previously been successfully used to estimate the area-averaged rain-rate distribution and the rainfall volume over an area from radar or from satellite infrared (IR) data. In most cases, the method was implemented over regions or test areas with an assumed homogeneous climatic character, that is, without a strong spatial variation of the rain regime throughout the test area. In the present paper, the behavior of the ATI method is discussed for a test area displaying two strong gradients of the cumulative annual rainfall: one meridional, at the transition between regions having, respectively, a desertic and an equatorial climate and the other zonal, at the transition between land and sea. The studied area is divided into four subtest areas (north, south, land, and sea) over which the ATI computation is applied separately. The linear coefficient relating the radar-observed area-averaged rain rate and the fractional area where the rain is higher than a threshold calculated over the four subtest areas is found to be almost constant, in agreement with the ergodic character of the rain-rate distribution observed in this region. Similarly, the linear coefficient relating the rain volume over the subtest areas to the IR satellite-derived ATI, a parameter analogous to the Geostationary Operational Environmental Satellite (GOES) Precipitation Index (GPI), is found to be very steady, with a mean value of 3.02 mm h−1 and a coefficient of variation of only 8%. These coefficients, as well as the underlying dynamic and microphysical processes, do not seem significantly influenced by the climatic character, even at a short space scale, in the studied area. The ratio of radar rain areas to cloud areas is, notably, almost constant. For a brightness temperature of 235 K, the ratio of the cloud area to rain area is around 1.68.

Exploratory analysis of the effect of hail suppression operations on precipitation in Alberta

An operational hail suppression program has been based in southern Alberta, Canada, since 1996. The program is designed to reduce hail damage to property in towns and cities. The analysis of the effect of the cloud seeding on the rainfall was motivated by concerns that hail suppression operations might reduce rainfall and thus offset any economic gain offered by a reduction of hail damage. An exploratory analysis of volume-scan, C-band radar data using sophisticated storm cell tracking software was used to calculate radar-derived rainfall characteristics from 160 seeded and 1167 non-seeded storms, on 82 days with seeding, during the summers of 2001 and 2002. The seeded storms (stratified according to maximum radar-derived cell top height) have greater mean durations (+50%), have greater mean precipitation rates (+29%) and have a greater mean total rain area–time integral (+54%). There is statistical evidence to reject the null hypothesis of no effect of cloud seeding on the total volume of rainfall. The data support the claim that seeding causes an increase in rainfall. The seeding effect is estimated to increase the mean rainfall volume (averaged for categories 7.5–11.5-km height storms) by a factor of 2.2, with an average 95% confidence interval of (1.4, 3.4).

Estimation of midlatitude rainfall parameters from satellite microwave radiometers using the area‐time integral concept

Weather Surveillance Radar‐1988 Doppler (WSR‐88D) level II data and special sensor microwave imager (SSM/I) brightness temperature data were collocated to further investigate the use of the area‐time integral (ATI) technique in the estimation of satellite microwave rainfall amount based on the storm area coverage information. The concept of a strong relationship between the areal extent and time duration of precipitation and the total amount of precipitation seems to have been extant in radar meteorology for some time, primarily in the tropics. This strong relationship is again demonstrated using WSR‐88D data. However, it was found that this correlation strongly depends on the Z‐R relationship used in computing rain rate from radar reflectivity data. This study conducted two experiments: (1) using one Z‐R relationship for the whole storm rain area and (2) separating rain into two parts, convective and stratiform, and applying two Z‐R relationships. The area‐wide average rain rate from the storm as viewed from the satellite was calculated from the ATI formalism based on the relationships developed from the WSR‐88D data. The estimates were compared with the SSM/I 85‐GHz scattering algorithm estimates and verified using Next Generation Weather Radar (NEXRAD) observations and hourly rain gauge measurements. The results reveal that the ATI technique can be used as an alternative approach for determining and validating satellite precipitation parameters. The separation of convective and stratiform precipitation improved the relationship between the area‐wide average rain rate and fractional rainfall area and consequently improved the accuracy of satellite rainfall estimation.

Multiparameter Radar Study of Rainfall: Potential Application to Area–Time Integral Studies

Multiparameter radars measure one or more additional parameters in addition to the coventional reflectivity factor. The combination of radar observations from a multiparameter radar is used to study the time evolution of rainstorms. A technique is presented to self-consistently compare the area-time integral (ATI) and rainfall volume estimates from convective storms, using two different measurements from a multiparameter radar. Rainfall volumes for the lifetime of individual storms are computed using the reflectivity at S band (10-cm wavelength) as well as one-way specific attenuation at X band (3-cm wavelength). Area-time integrals are computed by summing all areas in each radar snapshot having reflectivities (S band) in excess of a preselected threshold. The multiparameter radar data used in this study were acquired by the National Center for Atmospheric Research (NCAR) CP-2 radar during the Cooperative Huntsville Meteorological Experiment (COHMEX) and the Convection and Precipitation/Electrification Experiment (CaPE), respectively. ATI studies were accomplished in this work using multiparameter radar data acquired during the lifetime of six convective events that occurred in the COHMEX radar coverage area. A case study from the COMHEX field campaign (20 July 1986) was selected to depict the various stages in the evolution of a storm over which the ATI and rainfall volume computations were performed using multiparameter radar data. Another case study from the CaPE field campaign (12 August 1991) was used to demonstrate the evolution of a convective cell based on differential reflectivity observations.

Rainfall measurement by radar: a review

Several measuring methods using various properties of radar signals to estimate rainfall rate R are reviewed. These methods are based on measurements of the radar reflectivity factor Z at one or two polarizations, on measurements of attenuation A or on a combination of these two quantities. When only one radar observable is measured, R is computed from analytical relations of R- Z or A- Z type, ignoring the spatiotemporal variations of the raindrop size distribution N( D), where D is the drop diameter. If N( D) can be approximated by a two parameter equation, R can be estimated from the determination of N( D), using measurements of two radar observables. Some of these methods are notably improved if the radar data are combined with rain gauge network measurements at the ground. The mean volume rainfall and the areally averaged rain rate can be estimated from the area-time integral and fractional area methods.

The Relationship between Area–Time Integrals Determined from Satellite Infrared Data by Means of a Fixed-Threshold Approach and Convective Rainfall Volumes

The relationship of the rainfall from convective clouds to area-time integrals determined from satellite infrared data using a fixed infrared-temperature threshold is investigated. Concurrent radar and rapid-scan satellite data obtained during field projects in the northern High Plains and the southeastern United States were used in this study. The fixed IR threshold appropriate for each region was determined by an optimization procedure that identified the brightness threshold that yields the strongest relationship between estimated rainfall from a cloud cluster and its satellite Area-Time Integral (ATI) for each dataset. For the North Dakota-Montana area the optimization procedure indicated that the area enclosed by the -22.5 C isotherm provides satellite ATI values most closely related to the estimated rainfalls. For the southeastern United States project, the optimized tem- perature threshold was 8.5'C. The difference between the thresholds determined for the two geographic areas suggests that a different 'calibration' for each distinct area may be needed to make use of this relationship. Slopes of the two log-log rainfall-ATI regressions are less than unity, indicating that a relative horizontal expansion and/or increase in persistence of a cloud cluster exceeds the associated increase in precipitation. Implications for the Geostationary Operational Environmental Satellite precipitation index are discussed. New results concerning the rain volume-radar ATI relationship for the southeastern United States are also appended to the paper.

The Relation of Radar to Cloud Area-Time Integrals and Implications for Rain Measurements from Space

The relationships between satellite-based and radar-measured area-time integrals (ATI) for convective storms are determined, and both are shown to depend on the climatological conditional mean rain rate and the ratio of the measured cloud area to the actual rain area of the storms. The GOES precipitation index of Arkin (1986) for convective storms, an area-time integral for satellite cloud areas, is shown to be related to the ATI for radar-observed rain areas. The quality of GPI-based rainfall estimates depends on how well the cloud area is related to the rain area and the size of the sampling domain. It is also noted that the use of a GOES cloud ATI in conjunction with the radar area-time integral will improve the accuracy of rainfall estimates and allow such estimates to be made in much smaller space-time domains than the 1-month and 5-deg boxes anticipated for the Tropical Rainfall Measuring Mission.

The estimation of convective rainfall by area integrals: 1. The theoretical and empirical basis

This work develops a unified theory for the estimation of both total rainfall from an individual convective storm over its lifetime and the areawide instantaneous rainfall from a multiplicity of such storms by use of measurements of the areal coverage of the storms within a threshold rain intensity isopleth or the equivalent threshold radar reflectivity. The method is based upon the existence of a well‐behaved probability density function (pdf) of rain rate either from the many storms at one instant or from a single storm during its life. In the first method, the lifetime storm rainfall volume is V = [〈A(τ)〉T]S(τ), where 〈A(τ)〉 is the average storm area over the life of the storm of duration, T, in which R > τ and the bracketed term is the area time integral. In the second method, the instantaneous areawide rain rate, 〈R〉 = F(τ)S(τ), where F(τ) is the fractional observed area with R > τ. In both methods, S(τ) is the climatological rain rate for the regime divided by the relative frequency with which R > τ. For thresholds exceeding some minimum value, S(τ) is essentially linear with τ for the kind of lognormal pdf which characterizes convective rain, and is a constant for specified τ. Thus both the lifetime V of the individual storm and the instantaneous 〈R〉 for a multiplicity of storms are linear functions of A(τ) and F(τ), respectively. This is in accord with the observations of Doneaud et al. (1984) and Chiu (1988a, b). Because the autocorrelation time of the areawide rain rate of convective storms in areas in excess of 104 km2 is about 5–6 hours, the snapshot 〈R〉 is representative of the rain for a few hours. This enhances the accuracy of snapshot measurements for climate purposes and also extends their utility to smaller time/space problems such as hydrology and numerical weather prediction. We also discuss ways of obtaining climatologically valid and unbiased values of S(τ) and means of establishing its values from aircraft and space.

Estimation of Areal Rainfall Using the Radar Echo Area Time Integral

The Area Time Integral (ATI) method of Doneaud et al. (1984) is extended to the measurement of cumulative areawide rainfall for periods up to 12 h. The extended ATI method is used to analyze data from the Florida Area Cumulus Experiment II. The relationship between radar estimated rain volume and radar-measured area covered with echoes is examined to test the possibility of obtaining values similar to conventional reflectivity-rainfall estimates of rainfall using only area measurements. The correlation between gage-estimated rain volume and radar estimated area covered with showers is also studied, focusing on the possible estimation of rain volume values using a small number of echo area observations.

A Modified ATI Technique for Nowcasting Convective Rain Volumes over Areas

This paper explores the applicability of the area-time-integral (ATI) technique for the estimation of the growth portion only of a convective storm (while the rain volume is computed using the entire life history of the event) and for nowcasting the total rain volume of a convective system at the stage of its maximum development. For these purposes, the ATIs were computed from the digital radar data (for 1981-1982) from the North Dakota Cloud Modification Project, using the maximum echo area (ATIA) no less than 25 dBz, the maximum reflectivity, and the maximum echo height as the end of the growth portion of the convective event. Linear regression analysis demonstrated that correlations between total rain volume or the maximum rain volume versus ATIA were the strongest. The uncertainties obtained were comparable to the uncertainties which typically occur in rain volume estimates obtained from radar data employing Z-R conversion followed by space and time integration. This demonstrates that the total rain volume of a storm can be nowcasted at its maximum stage of development.

The Area-Time-Integral Technique to Estimate Convective Rain Volumes over Areas Applied to Satellite Data—A Preliminary Investigation

Early work attempting to apply GOES rapid scan satellite data to a recently developed simple radar technique used to estimate convective rain volumes over areas in a semiarid environment (the northern Great Plains) is described. Called the Area-Time-Integral (ATI) technique, it provides a means of estimating total rain volumes over fixed and floating target areas. The basis of the method is the existence of a strong correlation between the radar echo area coverage integrated over the lifetime of the storm and the radar estimated rain volume. The technique does not require the consideration of the structure of the radar intensities to generate rain volumes. but only the area covered by radar echoes. This fact might reduce the source of errors generated by the structure differences between the radar and the satellite signatures above given thresholds. Satellite and radar data from the 1981 Cooperative Convective Precipitation Experiment (CCOPE) and the North Dakota Cloud Modification Project (NDCMP) are used. Consecutive time steps with both radar reflectivities and satellite (VIS and IR) rapid wan data were considered during the evolution of six convective clusters: three on 12 June, and three on 2 July 1981. Radar echoes with reflectivity values ≥ 25 dBZ were used to define the area of rainfall and the respective digital unit thresholds within the satellite data delineating the rainy part of the cloud area. Correlation of the ATI versus IR digital count values was obtained for every time step and for the storm lifetime, respectively. A comparison of the stepwise evolution of radar parameters such as echo areas maximum echo heights, maximum reflectivities and satellite parameters such as threshold count values and coldest cloud top temperature is presented graphically and reflects the multicell characteristics of the convective clusters. Also, a comparison of radar and satellite parameters for the cluster lifetime is made. Satellite parameters pertaining to the cluster lifetime were derived both dependently and independently of radar data. The main purpose of this investigation is to compute convective rain volume of a convective cluster by application of the ATI technique based only on satellite data. As such, the key element is to determine the ATI from satellite data without consideration of radar data. This is possible if trends of satellite products generated independently are similar to those of satellite products based upon radar observations as done here. A parallel with the two-step techniques generally used to estimate rain volume from satellite data is made. To delineate the rainy part of a cloud area, a regression analysis is used. The regression relate a satellite-independent product to a satellite-dependent product. For a given storm. the satellite-independent product is first computed; then the regression equation gives the ATI, Finally, the rain volume is obtained by using the ATI versus rain volume relationship. By applying the ATI/rain volume relationship to satellite data, the errors generated by the complicated multiple area-volume transform relations am reduced, as similar errors were reduced when the technique was applied to radar data. In addition, a regression analysis gives more accurate estimates than a single threshold when used to delineate an area covered by rain events from an area covered by clouds. The advantages of the ATI technique are based on the fact that the technique operates on a storm lifetime integrated basis, while the previous techniques operate on a time-step basis. The new technique generates only total rain volume estimates (not rain rates). This indeed is a limitation. The analyses of six convective clusters suggest that the extension of the ATI technique using satellite data holds promise.

Convective Rain Rates and their Evolution during Storms in a Semiarid Climate

Rain rates and their evolution during summertime convective storms were analyzed for the semiarid climate of the northern High Plains. Radar data from a total of 750 radar echo clusters from the 1980 and 1981 summer cloud seeding operations of the North Dakota Cloud Modification project (NDCMP) were used. The analysis suggests that the average rain rate R¯ among storm is, in a first approximation, independent of the total rain volume if the entire storm duration is considered in the averaging process. This average rain rate depends primarily on the reflectivity threshold considered in calculating the area coverage integrated over the lifetime of the storm, the storm, the area-time integral (ATI). For the 25 dBz reflectivity threshold used in the ATI computations, R¯ was 4.0 mm h−1 with a standard deviation of 1.55 mm h−1, being ∼20% higher for wet mason conditions. The evolution of rain rates during storms was analyzed by dividing each storm lifetime into 10 min. 1, 2 and 4 h, and growing and decaying periods. A 10 min time increment was used in computing the parameters for all time intervals. A storm cluster reached its maximum growth after an average of 56% of its lifetime. The average rain rate for the growing period exceeded that for the decaying period by about 10%. As the time interval used in computations approached the storm lifetime, the scatter of the average rain rates was reduced, thus increasing the accuracy of rainfall estimates using the area time integral. The value of R¯ remained independent of the total rain volume when the growing or decaying periods of storms were considered separately. The total rain volume was also well correlated with the maximum single-scan rain volume. These findings suggest the possibility of estimating total storm rain volume at its maximum stage of development. It is hoped that improvements in rainfall estimation over area using satellite data may result from further studies since the precipitating part of a cloud picture can be more accurately defined for the growing period of a cloud's history.

The Area-Time Integral as an Indicator for Convective Rain Volumes

Abstract Digital radar data are used to investigate further a simple technique for estimating rainfall amounts on the basis of area coverage information. The basis of the technique is the existence of a strong correlation between a measure of the rain area coverage and duration called the Area-Time Integral (ATI) and the rain volume. This strong correlation is again demonstrated using echo cluster data from the North Dakota Cloud Modification Project 5 cm radars. Integration on a scan-by-scan basis proved to be superior for determining ATI values to the hour-by-hour integration used previously. A 25 dB(z) reflectivity threshold was found suitable for the ATI calculation. The correlation coefficient on log-log plots of cluster rain volume versus ATI is approximately 0.98, indicating a power-law relationship between the variables. The exponent of that relationship is just a little higher than one, which indicates that the cluster average rainfall rate is almost independent of the storm size and duration. A ...