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  • View in gallery

    Time–height cross section of (a) reflectivity, (b) mean Doppler velocity, and (c) spectrum variance from 920-MHz profiler for 1000 UTC 23 Jan–0000 24 Jan 2006.

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    Doppler velocity spectra from (a) 50- and (b) 920-MHz profiler. Air motion is estimated from 50-MHz profiler and plotted as asterisks on 920-MHz profiler spectra. DSD is estimated from 920-MHz profiler spectra that observe Doppler motion of hydrometeors.

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    Evolution of retrieved integral parameters below the bright band (at h = 4000 m) for the rain event observed from 2000 UTC 23 Jan to 0000 24 Jan 2006 in Darwin, Australia: (a) Z, (b) RNT, (c) LWC, and (d) M0.

  • View in gallery

    Evolution of the retrieved parameters of the modified Gamma DSD below the bright band (at h = 4000 m). (bottom) The median drop diameter (d0) and the shape factor (μ). (top)The corresponding DSDs grouped by (left to right) hours of the rain event: 2000–2100, 2100–2200, 2200–2300, and 2300–0000 UTC.

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    Comparison between modeled (M) and retrieved (R) vertical parameters for (left to right) 2213, 2233, and 2253 UTC for (a) Z, (b) RNT, and (c) LWC. The residual: Res. = (MR)/R is displayed in the last column. Notation: 22.XX = 22XX UTC.

  • View in gallery

    Evolution of (left) Z, (middle) RNT, and (right) LWC for (top to bottom) different fall distances (Dfall)—from 100 to 3500 m—from the top of the rain shaft [i.e., height (h) from ground level] for 2000–0000 UTC: model results and retrieved data.

  • View in gallery

    Scatterplots for (left) Z, (middle) RNT, and (right) LWC for model results and radar data for different fall distances (Dfall)—from 100 to 2500 m—from the top of the rain shaft [i.e., height (h) from ground level] for 2000–0000 UTC.

  • View in gallery

    Values of the slope of the DSD parameter Λ for randomly generated exponential DSDs for the period 2000–0000 UTC.

  • View in gallery

    Comparison of vertical profiles from model (randomly generated MP DSD) and retrieved (TWP-ICE) at the top of the rain shaft (h = 4000 m) for the period 2000–0000 UTC for (a) Z, (b) RNT, and (c) LWC.

  • View in gallery

    Comparison of the evolution of integral parameters retrieved from the JWD disdrometer at ground level) with model results); percent difference between model and JWD: (a) Z, (b) RNT, and (c) LWC.

  • View in gallery

    Comparison of DSD at ground level between model (times signs) and JWD (plus signs) at (a) 2020, (b) 2040, (c) 2140, (d) 2120, (e) 2220, (f) 2240, (g) 2320, and (h) 2340 UTC.

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    Comparison of M0 for 2000–0000 UTC between model and JWD. For modeling results, integral parameters are computed using a summation on all 40 bins and a summation on bins corresponding to JWD channels for which drops are detected.

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An Intercomparison of Model Simulations and VPR Estimates of the Vertical Structure of Warm Stratiform Rainfall during TWP-ICE

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  • 1 Civil and Environmental Engineering Department, Pratt School of Engineering, Duke University, Durham, North Carolina
  • | 2 Cooperative Institute for Research in Environmental Sciences, University of Colorado, and NOAA/Earth System Research Laboratory, Boulder, Colorado
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Abstract

A model of rain shaft microphysics that solves the stochastic advection–coalescence–breakup equation in an atmospheric column was used to simulate the evolution of a stratiform rainfall event during the Tropical Warm Pool-International Cloud Experiment (TWP-ICE) in Darwin, Australia. For the first time, a dynamic simulation of the evolution of the drop spectra within a one-dimensional rain shaft is performed using realistic boundary conditions retrieved from real rain events. Droplet size distribution (DSD) retrieved from vertically pointing radar (VPR) measurements are sequentially imposed at the top of the rain shaft as boundary conditions to emulate a realistic rain event. Time series of model profiles of integral parameters such as reflectivity, rain rate, and liquid water content were subsequently compared with estimates retrieved from vertically pointing radars and Joss–Waldvogel disdrometer (JWD) observations. Results obtained are within the VPR retrieval uncertainty estimates. Besides evaluating the model’s ability to capture the dynamical evolution of the DSD within the rain shaft, a case study was conducted to assess the potential use of the model as a physically based interpolator to improve radar retrieval at low levels in the atmosphere. Numerical results showed that relative improvements on the order of 90% in the estimation of rain rate and liquid water content can be achieved close to the ground where the VPR estimates are less reliable. These findings raise important questions with regard to the importance of bin resolution and the lack of sensitivity for small raindrop size (<0.03 cm) in the interpretation of JWD data, and the implications of using disdrometer data to calibrate radar algorithms.

Corresponding author address: Dr. Ana P. Barros, Duke University, Box 90287, 2457 CIEMAS Fitzpatrick Bldg., Durham, NC 27708. Email: barros@duke.edu

Abstract

A model of rain shaft microphysics that solves the stochastic advection–coalescence–breakup equation in an atmospheric column was used to simulate the evolution of a stratiform rainfall event during the Tropical Warm Pool-International Cloud Experiment (TWP-ICE) in Darwin, Australia. For the first time, a dynamic simulation of the evolution of the drop spectra within a one-dimensional rain shaft is performed using realistic boundary conditions retrieved from real rain events. Droplet size distribution (DSD) retrieved from vertically pointing radar (VPR) measurements are sequentially imposed at the top of the rain shaft as boundary conditions to emulate a realistic rain event. Time series of model profiles of integral parameters such as reflectivity, rain rate, and liquid water content were subsequently compared with estimates retrieved from vertically pointing radars and Joss–Waldvogel disdrometer (JWD) observations. Results obtained are within the VPR retrieval uncertainty estimates. Besides evaluating the model’s ability to capture the dynamical evolution of the DSD within the rain shaft, a case study was conducted to assess the potential use of the model as a physically based interpolator to improve radar retrieval at low levels in the atmosphere. Numerical results showed that relative improvements on the order of 90% in the estimation of rain rate and liquid water content can be achieved close to the ground where the VPR estimates are less reliable. These findings raise important questions with regard to the importance of bin resolution and the lack of sensitivity for small raindrop size (<0.03 cm) in the interpretation of JWD data, and the implications of using disdrometer data to calibrate radar algorithms.

Corresponding author address: Dr. Ana P. Barros, Duke University, Box 90287, 2457 CIEMAS Fitzpatrick Bldg., Durham, NC 27708. Email: barros@duke.edu

1. Introduction

Many attempts have been made over the years to describe the evolution of rainfall microstructure in the presence of coalescence, breakup, accretion, evaporation, and condensation mechanisms (Testik and Barros 2007). This research lead to the development of several one-dimensional homogeneous models for the prediction of the temporal evolution of the vertical structure of warm rain in the presence of coalescence–breakup (List et al. 1987, hereinafter LDS87; Tzivion et al. 1989, hereinafter TFL89; List and McFarquhar 1990, hereinafter LMF90; McFarquhar and List 1991, hereinafter MFL91; Feingold et al. 1991, hereinafter FLT91; Hu and Srivastava 1995, hereinafter HS95; Prat and Barros 2007a, b), evaporation–condensation (TFL89; FLT91; HS95), surrounding air motion (FLT91; HS95), and unsteady rain boundary conditions (LDS87; LMF90; MFL91). From a general point of view, such models proved able to predict qualitatively the evolution of the integral parameters of the droplet size distribution (DSD) or the evolution of the DSD with increasing fall distance within an atmospheric column. However, one shortcoming of previous studies concerns the lack of, or limited evaluation of models against observations. Typically, comparisons were performed using a limited amount of experimental data (DSDs) mostly obtained with disdrometer measurements (LMF90; MFL91; HS95). Nevertheless, disdrometers that were initially designed to measure drop size distributions in order to calculate radar reflectivities (Sheppard 1990) only provide measurements at the ground level, and therefore they cannot be used to infer the spatial and temporal evolution of the vertical profiles of the DSD and its associated integral parameters, such as reflectivity (Z), rain rate (RNT), and liquid water content (LWC). A previous study by McFarquhar et al. (1996) presented measurements of pulsating rain obtained with a ground-based disdrometer with a scanning X-band Doppler radar operating at 9375 MHz. A comparison with earlier modeling results (LMF90; MFL91) provided qualitative agreement regarding the pulsating nature of rain and the transient reflectivity patterns observed at ground level. However, model simulations that were limited to the use of a constant rain rate at the top of the rain shaft did not provide a quantitative agreement with observed varying rainfall intensity.

Vertically pointing radars (VPRs) have been used over two decades and have proved useful for retrieval of the vertical structure of the raindrop size distribution (see Wakasugi et al. 1986; Rajopadhyaya et al. 1999; Williams 2002 for key references). The combination of several profilers (usually two) with different operating frequencies [usually very high frequency (VHF) and ultrahigh frequency (UHF)], is able to capture distinct characteristics of the vertical structure of precipitation, and allows a detailed reconstruction of the rain event. Profiler observations at long wavelengths (usually 50 MHz) are used to estimate the ambient air motion and turbulent broadening in the finite radar beam as the precipitation passes directly over the field site. The shorter wavelength observations (typically 915, 920, or 2875 MHz) are used to estimate the DSD throughout the atmosphere. However, depending on the operating frequency, one of the caveats of vertically pointing radars is their difficulty in measuring precipitation in the lower part of the troposphere, typically for altitudes below 1000 m.

Previously, our research efforts focused on developing a numerical model of the evolution of the rainfall microstructure in warm rain in presence of coalescence–breakup mechanisms (Prat and Barros 2007a, b). In this work, a dynamic simulation of the evolution of the drop spectra within a one-dimensional rain shaft is performed using realistic boundary conditions retrieved from real rain events. DSDs retrieved from vertically pointing radar measurements are sequentially imposed at the top of the rain shaft as boundary conditions in order to emulate a realistic rain event. Subsequently, the simulated temporal evolution of integral parameters (i.e., Z, RNT, and LWC) is compared with radar-based retrievals at different heights throughout the rain shaft and against ground-based disdrometer observations. Finally, we demonstrate the benefit of using the model as a physical interpolator to improve radar rainfall estimation at low levels in the atmosphere, and specifically between the cloud base and the ground surface.

The paper is organized as follows. The rain shaft model is briefly described in section 2, including the strategy for the model–data comparison, the framework used for incorporating the field data into the model, and a description of the real rainfall dataset used for comparison. Model results are compared with the Tropical Warm Pool-International Cloud Experiment (TWP-ICE) data in section 3. This section consists of two parts. First, results obtained with the rain shaft model are compared with VPR data at different heights throughout the atmospheric column. Second, the model’s ability to predict the evolution of rain microstructure near the surface is evaluated against the Joss–Waldvogel disdrometer (JWD) data. Last, a strategy to achieve a reliable dynamic simulation of the DSD is discussed.

2. Model description

The approach presented in this work could have been performed with any bin model. We chose to use a number and mass conservative bin model with state-of-the art collision and breakup kernels, which has fundamental advantages. We will briefly describe the key components of the one-dimensional homogeneous rain shaft model (Prat and Barros 2007a), but the interested reader is referred to Prat and Barros (2007a, b) for more information. The general continuous transport equation is discretized using a grid independent number and mass conservative scheme (Kumar and Ramkrishna 1996). The resulting discrete equation over the ith discrete interval is given by (Prat and Barros 2007a):
i1558-8432-47-11-2797-e1
where Ni(z, t) is the total number density of droplets in the ith class size (cm−3):
i1558-8432-47-11-2797-e2

The terms on the right-hand side of Eq. (1) describe the coalescence–breakup dynamics with the rate of change due to gains by coalescence (first term), losses by coalescence (second term), gains by breakup (third term), and losses by breakup (fourth term). The term (xi) is the representative volume of the ith size range. The collection kernel (Ci,j) is the product of the coalescence efficiency (Ecoali,j: Low and List 1982a) and the gravitational collision kernel (Ki,j: Pruppacher and Klett 1978). The breakup (Bi,j) and collection kernels are complementary (i.e., Bi,j = 1 − Ci,j). The parameters η (coalescence contribution) and κi,j,k (breakup contribution) are derived from the number-mass conservative discretization scheme (Prat and Barros 2007b). Specifically, η is the contribution to the droplet population located at the ith interval due to coalescence of drops of volume (xj) and (xk). When a drop of volume (υ = xj + xk) is created by coalescence and its size falls between two grid points (xi) and (xi+1), the newly created drop is distributed proportionally to both adjacent grids points (xi and xi+1) in such a way that both number and mass are preserved. Similarly, the term κi,j,k is the contribution to droplet population located at the ith interval due to collisional breakup of droplets of volume (xj) and (xk). The parameter (κi,j,k) is a function of the fragment distribution function P(υ, xj, xk), where P(υ, xj, xk) is the number of drops with volume in the range (υ) to (υ + ) obtained from the collisional breakup of two drops of volume (xj) and (xk) (Prat and Barros 2007a). The overall fragment distribution function include the three types of breakup (filament: FI, sheet: SH, disc: DI) identified by McTaggart-Cowan and List (1975). The occurrence (ratio) of each type of breakup (RXX with XX = FI, SH, or DI) and the number of drops (FXX with XX = FI, SH, or DI) created by a collisional event is provided by Low and List (1982b). The fragment distribution function P(υ, xj, xk) for each type of breakup is derived from the parameterization proposed by McFarquhar (2004a) that provides analytical expressions for the parameters (height: H; standard deviation: σ; modal diameter: μ) of the Gaussian and lognormal functions constituting the fragment distribution for each type of breakup. The term κi,j,k is analytically integrated over every bin category via routines specifically written for generalized Gaussian and lognormal distributions (Prat and Barros 2007a). The overall fragment distribution is obtained from the weighted contribution of each type of breakup. As pointed out previously (McFarquhar 2004a; Prat and Barros 2007a, b), this formulation (McFarquhar 2004a) has the double advantage of having a more consistent physical basis in order to generalize experimental results of collisional breakup to arbitrarily selected pairs of colliding drops, thus overcoming the challenge posed by the original Low and List parameterization (1982a, b) regarding the conservation of mass for each breakup event. Moreover, this formulation (McFarquhar 2004a) is able to predict the breakup function for pairs of colliding drops, other than the original pairs reported in Low and List (1982a, b), when compared with laboratory experiments carried out at National Aeronautics and Space Administration’s (NASA’s) Wallops Island Facility (Barros et al. 2008).

The irregular discretization grid covers a diameter range from 0.01 to 0.7 cm (nbin = 40 bins), and combines a geometric grid for (d ≤ 0.1cm) to capture accurately the location of peaks in the small drop diameter range with a regular grid (Δd = 0.02 cm) for (d ≥ 0.1 cm) to capture accurately the right-hand-side tail of the DSD and minimize numerical diffusivity (Prat and Barros 2007a, b). The moments of the DSD are expressed by
i1558-8432-47-11-2797-e3
where mi is the characteristic mass (g) of the ith class category. The term CC is a conversion coefficient that depends on the integral properties considered. Moments of interest include, the drop number concentration M0(z, t) (k = 0; CC = 1; cm−3), the liquid water content LWC(z, t) (k = 1; CC = 106; g m−3), and the radar reflectivity factor Zmm6m−3 (z, t) {k = 2; CC = 1012[6/(π ρ)]2 with ρ = 1 g cm−3; mm6 m−3 or dBZ} with Z(z, t) = 10 logZmm6m−3 (z, t). Finally, the rain rate [RNT(z, t), mm h−1] is given by
i1558-8432-47-11-2797-e4
with CC = 3600/ρ, and Vi(z) = [ViVair(z)], where Vi (m s−1) is the drop fall velocity for a drop of the ith bin category (Atlas et al. 1973) and Vair(z) is the mean air velocity at the height (z) that includes updrafts–downdrafts effects with appropriate sign conventions (+for downdraft and −for updraft).

The numerical approximation of Eq. (1) is achieved using a semi-Lagrangian formulation with a vertical resolution Δz = 10 m and a time step Δt = 1 s. At each time step the distribution of drops inside the rain shaft is updated using the vertical streamlines for each drop size class (Lagrangian reference), and the DSD spectra is then modified locally under the influence of combined coalescence–breakup mechanisms (Eulerian reference).

3. Approach to integration of model and field observations

a. Description of the rainfall dataset used for comparison

Vertically pointing profiling radar observations were collected during the TWP-ICE around Darwin, Australia, in January and February 2006. The experiment provided both remote sensing observations and aircraft in situ measurements within anvil clouds needed to verify the remote sensing microphysical properties needed to examine the life cycle of convective cloud systems from their initial forming stage to the decaying of thin, high-level cirrus. For this study, the Doppler velocity spectra collected by the collocated 50- and 920-MHz profiling radars were used to estimate the vertical air motion and the vertical profile of rain DSDs. The long wavelength 50-MHz profiler observations are used to estimate the ambient air motion and turbulent broadening in the finite radar beam as the precipitation passes directly over the profiler site. The shorter wavelength 920-MHz profiler observations are used to estimate the DSD. The two profilers, along with a JWD and two tipping-bucket rain gauges were deployed at the Bureau of Meteorology Research Center (BMRC) wind profiler site located at −12.44° and 130.96°.

The time–height cross sections of reflectivity, mean Doppler velocity, and spectral width of the precipitation event studied are shown in Fig. 1. After approximately 1930 UTC, the radar brightband signature can be seen in the reflectivity observations indicating stratiform rain. Another indicator of stratiform rain is the transition from snow and ice falling at 1–2 m s−1 above h = 5 km into raindrops falling at 5–8 m s−1 below h = 5 km. A third indicator of stratiform rain is the decrease in spectral width above the snow–rain transition altitude indicating that turbulent motions associated with convective and transitional precipitation processes have diminished at high altitudes directly over the profiler site. This rain event occurred during the active phase of the monsoon with strong low-level easterly winds transporting moisture off of the Timor Sea. The rain system consisted of large areas of stratiform rain with embedded convection. Two convective cells passed over the profiler site near 1550 and 1820 UTC, and can be identified by the enhanced reflectivity and increased spectral width between h = 3 km and h = 6 km.

Examples of observations made by the two profilers while precipitation was directly over the profiler site are shown in Fig. 2. Figure 2a was derived from the 50-MHz profiler and Fig. 2b was derived from the 920-MHz profiler. The 920-MHz profiler reflectivity is shown in the right-hand panel of Fig. 2b with units of dBZ = 10 log10 mm6 m−3. The panels show the reflectivity-weighted Doppler velocity spectral density (often called just the “Doppler spectrum”) in units of 10 log10 mm6 m−3 (m s−1)−1 at each range gate. The logarithmic scale is used to aid in visualizing the data that spans six orders of magnitude.

The advantage of using the 920-MHz operating frequency radar is that the radar is sensitive to backscattered energy from hard targets distributed throughout the radar pulse volume. Referring to Fig. 2b, the 920-MHz profiler Doppler spectra show particles with net downward motion below the freezing level, which is located around h = 4.5 km. The solid black lines in the panels indicate the air–density-corrected terminal fall speeds of raindrops with diameters of 1, 3, and 6 mm.

b. DSD retrieval procedure

Here we give a few elements on the retrieval procedure using VPR. For more details concerning the methodology, the reader is referred to Rajopadhyaya et al. (1999), Schafer et al. (2002), Williams (2002), Lucas et al. (2004), and Williams et al. (2007). Assuming a perfect radar with infinitesimal beamwidth and a static atmosphere with zero vertical air motion and zero turbulence, the observed reflectivity-weighted Doppler spectral density, Shydrometeor(V), is uniquely related to the raindrop size distribution, N(d), through the relation (Atlas et al. 1973):
i1558-8432-47-11-2797-e5
where V and d(V) are the velocity and velocity resolution, respectively, of the Doppler spectrum (m s−1); and d and d(d) are the diameter and diameter resolution corresponding to each V and d(V), respectively (mm). Of course, d has positive definite values and d and d(d) are valid only when V corresponds to downward air-density-corrected raindrop terminal fall speeds. Through laboratory studies, the transformation from velocity to diameter space in Eq. (5) is facilitated with a raindrop diameter to terminal fall speed relationship expressed as
i1558-8432-47-11-2797-e6
where ρ0 and ρ represent the air densities at sea level (ρ0 = 1.225 kg m−3) and at the level of the observation aloft, respectively (Gunn and Kinzer 1949; Atlas et al. 1973). The Doppler velocity reflectivity spectral density observed by a radar is not just the simple expression of Eq. (5), but is the reflectivity spectral density shifted by the ambient air motion and broadened by the turbulent motions and horizontal wind motions within the radar pulse volume. For the 920-MHz profiling radar, the observed reflectivity spectral density, Smodel(V), can be expressed as
i1558-8432-47-11-2797-e7
where Sair(Vω, σair) is the spectral-shift and spectral-broadening function corresponding to the vertical air motion ω and total spectral broadening σair and the asterisk refers to the convolution function (Wakasugi et al. 1986).

In Fig. 2a, the Doppler spectra show that the 50-MHz profiler is sensitive to the backscattered energy from distributed hard targets (as is the 920-MHz profiler) and it is also sensitive to the backscattered energy from changes in the turbulent refractive index (Gage and Gossard 2003). This second sensitivity is observed in the panel near zero velocity from near the surface to about h = 5 km. The asterisks and horizontal line at each range gate indicate the vertical air motion and spectral broadening determined by a best fit to a Gaussian function following the method of Gossard (1994).

Using the vertical air motion ω and spectral broadening σair estimated from the 50-MHz profiler Doppler velocity spectrum, the raindrop size distribution N(d) can be retrieved from the 920-MHz profiler Doppler velocity spectrum using Eqs. (5)(7) and inverse modeling techniques that are dependent on the assumed raindrop size distribution. The raindrop size distribution N(d) is modeled using a normalized Gamma function DSD (Willis 1984; Illingworth and Blackman 2002; Bringi et al. 2004) as described in Williams and Gage (2008, manuscript submitted to Ann. Geophys.) where
i1558-8432-47-11-2797-e8
where N(d) (cm−4) is the raindrop size distribution, Nz (mm6 m−3) is the total reflectivity, μ is the shape factor, d0 (mm) is the profiler retrieved median drop diameter, and fz(d0, μ) is given by
i1558-8432-47-11-2797-e9

One benefit of describing N(d) using Eq. (8) is that Nz represents the reflectivity of the observation and is independent of d0 and μ, which determine the shape of the raindrop size distribution. The three parameters (Nz, μ, and d0) describing the normalized Gamma DSD are determined by iteratively adjusting the parameters until minimizing the sum of the squared difference between the modeled spectrum described by Eq. (7) and the observed 920-MHz profiler Doppler velocity spectrum. Whereas Nz was held constant at the measured reflectivity, all values of d0 between 0.25 and 3.5 mm at 0.01-mm steps and values of μ between −2 and 20 at unit steps were evaluated to find the global minimized solution. Estimates of N(d) are made at each vertical range gate of the radar independent of estimated obtained at other range gates. More specific details of inverse modeling techniques used to retrieve N(d) from profiling radar observations is given in Williams and Gage (2008, manuscript submitted to Ann. Geophys.).

c. Model incorporation of retrieved rainfall data

As pointed out earlier, the goal of this study is to achieve a dynamic simulation of a realistic warm rain event and compare the spatial and temporal evolution of integral parameters throughout the 1D rain shaft model with VPR measurements at different levels within the shaft. In previous studies, the unsteady behavior of rain was imposed by varying initial conditions at the top of the rain shaft generated synthetically using a sinusoidal DSD (LDS87), or single-pulse (LMF90) and multiple-pulse rain (MFL91) characteristics determined from ground-based disdrometer measurements. In this work, the 1-min time series average DSD retrieved from VPR measurements is imposed at the top of the rain shaft, and the evolution of the DSD spectra and related integral parameters are computed at each level throughout a 4-km rain shaft. In the stratiform rain event considered in this work (2000–2400 UTC), the bright band is clearly visible and located around h = 4.2–4.5 km as observed in Fig. 1. The selection of a 4-km rain shaft allows us to consider a rain event free of ice microphysics.

Figure 3 presents the evolution of the retrieved integral parameters below the freezing level (h = 4 km) for the 2000–0000 UTC period. Figure 3a displays the evolution of the radar reflectivity factor (Z). The period 1700–2000 UTC (not displayed here) is characterized by strong variations in the value of Z indicating strong convective activity with intense updrafts–downdrafts (respectively between ±5 m s−1) as indicated by the retrieved mean air velocity with the 50-MHz profiler (not shown). For the stratiform period 2000–0000 UTC, the radar reflectivity presents smoother variations than for the period 1700–2000 UTC along with little or moderate updrafts–downdraft (respectively between 0.5 m s−1; not shown). Other integral parameters of the DSD indicate a light/moderate rain event with rain rates between 0.7 and 7 mm h−1, and liquid water content between 0.03 and 1 g m−3. In addition, the drop number concentration (M0) computed from the retrieved DSD varies between 10−5 and 10−1 drops per centimeter cubed for the period 2000–0000 UTC.

In Fig. 4, the transient evolution of the retrieved parameters of the normalized gamma DSD: the median drop diameter (d0) and the shape parameter (μ) (parameter Nz not shown), indicates that the variability of the DSD decreases with time as a result from the transition from a convective to a stratiform rainfall regime. For this period, hourly averaged values of (d0) increase from 0.12, 0.145, 0.16, to 0.165 cm, respectively, for 2000–2100, 2100–2200, 2200–2300, and 2300–0000 UTC. Simultaneously, the standard deviation decreases from 0.039 (2000–2100 UTC), 0.020 (2100–2200 UTC) to 0.015 (2200–2300 UTC), and finally increases up to 0.019 (2300–0000 UTC). The trend for the shape factor (μ) is more erratic with hourly averaged values (and standard deviation) of 1.7 (1.99), 0.36 (1.92), 1.5 (1.36), and 2.22 (1.57) for each hourly interval, respectively. A similar conclusion can be drawn when looking at the hourly evolution of the shape of the minute-by-minute retrieved DSD of the stratiform rainfall. While profiler DSD estimates are available every 20 s, the interval of 1 min was selected in order to obtain stable and representative averaged DSDs and remove noisy short-term perturbations due to radar measurement uncertainties. These minute-averaged retrieved DSDs, updated every minute and held constant during each time step of the simulation were imposed as the top boundary condition for the rain shaft (i.e., at h = 4 km). Except for the first time step (t = 0) when the entire column is initialized with the same DSD, each level of the shaft (z) is not constrained by any external condition and is thus let free to evolve during the simulation according to settling-microphysics (coalescence–breakup) processes only.

4. Modeling results

a. Model comparison with radar rainfall data

Figure 5 displays a comparison for the vertical profiles of integral parameters (Z, RNT, and LWC) of the DSD at three different times (2213, 2233, and 2253 UTC) during the stratiform rain event sequence from 2000 to 0000 UTC. The model tracks the evolution of the vertical profiles of integral parameters, and capture the temporal evolution of the retrieved reflectivity profile with an average residual between retrieved and simulated Z of about 5% for the reflectivity expressed in reflectivity decibels [about 20%–30% when expressed in linear scale (mm6 m−3)] once effects of the uniform initial condition have vanished (not shown). For the RNT and LWC, differences between model results and VPR retrieved data are significantly higher than for Z [when expressed in linear scale (mm6 m−3)] and remain in the range 20%–50%. Some of the largest differences can be explained by the differences in the fit of the DSD and the large uncertainty in the retrieved vertical velocity estimates. In addition please note that Fig. 5 is here to provide an instantaneous representation of the vertical range of differences between modeled and retrieved profiles at arbitrarily selected times. As we will discuss later, the average residuals for the integral parameters between modeled and VPR retrieved time series are found to be within the range of experimental uncertainties.

The transient evolution of reflectivity, rain rate, and liquid water content for selected fall distances (Dfall = 100, 500, 1500, 2500, and 3500 m) from 2000 to 0000 UTC is analyzed in Fig. 6. Besides measurement uncertainty, differences observed are due to the relatively simplified formulation of the homogeneous shaft model that does not take into account the local variation of the flow field around falling drops, shear, and horizontal wind effects. However, despite this simplification, the difference between modeling results and retrieved data for Z remains within 5% (dBZ scale) for all cases presented. Simulated evolutions for rain rate and liquid water at specific heights present average differences around 25% against VPR retrieved values, with a value significantly higher for the liquid water content than for the rain rate. In addition, we note that for fall distances greater than Dfall = 3000 m (i.e., h = 1000 m AGL), large differences are observed between model and experiments from 2000 to 2400 UTC. Those differences are due to the spatial resolution of the 50-MHz VPR (h = 1500 m) that does not permit reliable estimation of the magnitude of air motion below this limit (Williams 2002). For the air parcel located between h = 0–1500 m AGL, the mean velocity is assumed constant at the retrieved air velocity at h = 1500 m. Values of the correlation coefficient for integral parameters (Z, RNT, LWC) between modeling results and retrieved data time series (Table 1) indicate that differences increase with increasing fall distance (Dfall) from the h = 4-km level at which retrieved DSDs were imposed as boundary conditions. For instance, modeled and retrieved time series of the reflectivity (Z) present a degree of correlation ranging from 0.99 close to the top of the rain shaft (Dfall = 100 m) to 0.83 at the bottom of the rain shaft (at Dfall = 3000 m) giving an average correlation of 0.92 throughout the rain shaft. Closer to the ground level (Dfall = 3500 m), the correlation between the two times series decreases sharply (0.67) due to the presence of aforementioned artifacts. Comparatively, correlation coefficients obtained for RNT and LWC time series are lower than for Z with an average value of 0.83 and 0.78 for (RNT) and (LWC), respectively, throughout the column. Unlike reflectivity (Z), for which the coefficient correlation remains high throughout the shaft, the correlation for LWC decreases significantly closer to the top of the shaft with a strong variation between fall distances Dfall = 100 m (0.94) and Dfall = 200 m (0.80). Again, as pointed out for (Z) close to ground level (Dfall = 3500 m), the correlation coefficients for RNT (0.06) and LWC (0.07) time series are very low due to the difficulties for a correct measurement of the vertical air motion with the 50-MHz profiler. Also note that the radar reflectivity (Z) estimate is independent of vertical air motion and is more accurate than RNT and LWC below h = 1500 m.

Likewise, scatterplots of reflectivity, rain rate, and liquid water content for modeling results and radar retrievals at different fall distances (Dfall = 100, 500, 1500, and 2500 m) show an increase in dispersion with increasing fall distance (Fig. 7). The linear regression analysis performed for the reflectivity gives a coefficient of determination R2 ranging from 0.93 (Dfall = 100 m) to 0.64 (Dfall = 3500 m: not shown). Results for the rain rate and the liquid water content are comparable with R2 ranging from 0.86 (Dfall = 100 m) to 0.64 (Dfall = 3000 m) and from 0.82 (Dfall = 100 m) to 0.68 (Dfall = 3500 m), respectively.

Overall, the results suggest that the 1D homogenous shaft model captures closely the evolution of integral properties Z, RNT, and LWC. However, a true deterministic comparison as performed above, with retrieved boundary conditions sequentially imposed at the input of the shaft, is delicate. This approach, in which DSDs are updated each minute, does not take into account small time-scale variations and detailed flow fields and shear effects on the drop motion. From a modeling point of view, the model itself is not free of numerical uncertainties due to approximations introduced by a simplified space–time–bin discretization scheme for the resolution of the general advection–coalescence–breakup transport equation as indicated by the progressive decrease of correlation and determination coefficients with increasing fall distance from the level (h = 4 km) taken as a reference for modeled and retrieved integral properties.

Another possible explanation to the increasing differences between modeled and VPR retrieved vertical profiles with increasing fall distance relates to the model quasi-stochastic assumption that uses an average breakup function (kernel) to describe the drop population resulting from drop–drop interactions. A quasi-stochastic behavior is truly valid for significant rain-rate intensities (50 mm h−1: McFarquhar 2004b), higher than those retrieved in the present study (0.7–7 mm h−1). A complementary study (not shown) in order to determine the mean free path of drops (average distance for which a drop of diameter d will experience a collision with any other drops) as per McFarquhar (2004b), was performed for all DSDs imposed at the top of the shaft. Briefly, simulations show that drops of a given diameter can experience between 1 and 0 (i.e., average mean free path of 4–10 km for drops around d = 0.1–0.15 cm) and 1000–100 (average mean free path of 4–40 m for drops above d = 0.3 cm) collisions during their 4-km fall (for all DSDs retrieved during the period 2000–0000 UTC) for rain rates between 0.7 and 7 mm h−1. Those results are comparable to those obtained by McFarquhar (2004b) for exponential [Eq. (10): Marshall and Palmer 1948, hereinafter MP48] and for normalized gamma (Willis 1984) DSDs. More importantly, this study (not shown) also indicated that the shape of the retrieved normalized gamma DSD (i.e., values of the parameters: Nz, μ, and d0) was of prime importance as it concerns the value of the mean free path. The rain-rate intensity does not appear to be the key parameter governing drop–drop interaction statistics as it is the case for MP48 DSDs, with decreasing mean free path for increasing rain rates (McFarquhar 2004b).

From an experimental perspective the radar retrieval algorithm assumes a priori a normalized gamma distribution [Eqs. (8) and (9)] as the best fit for the retrieved DSD. The use of a normalized gamma distribution (Willis 1984; Illingworth and Blackman 2002; Bringi et al. 2004) requires the determination of three parameters (Nz, μ, and d0) using the error minimization procedure described earlier, a challenge for occasional spectra when the inverse model converges to an unrealistic value of the shape factor μ. The use of other formulations such as the exponential [MP48; see Eq. (10)], gamma (Ulbrich 1983), and lognormal (Sauvageot and Lacaux 1995) distributions could address this problem in part as discussed in Williams and Gage (2008, manuscript submitted to Ann. Geophys.). By contrast, the model does not make assumptions concerning the shape of the DSD, and the evolution of the drop number concentration in each one of the 40-class category is controlled by exchanges between each class sizes from relative contributions of sedimentation–coalescence–breakup processes.

To assess the model sensitivity, ensemble simulations were performed using randomly generated DSDs from the statistical distribution derived from the reflectivity observations. Instead of specifying a normalized gamma DSD as the top boundary condition [Eqs. (8) and (9)], for simplicity we use a variable exponential DSD [Eq. (10)] that requires generating only one random variable (slope: Λ) for each exponential distribution rather than two (or three) nonlinearly related parameters in case of more complex formulations (i.e., gamma, or normalized gamma):
i1558-8432-47-11-2797-e10

For this exercise, 60 1-min exponential DSDs were randomly generated to be used as top of the model boundary conditions. Figure 8 displays the randomly generated exponential DSD slope parameters (Λ) bounded by the observed hourly upper (Λmax) and lower (Λmin) slope parameters that were determined from the retrieved normalized gamma DSD (Fig. 4). Values of the constant (N0), similarly determined from retrieved normalized gamma DSD, were held equal to N0 = 0.08 cm−4 (2000–2100 UTC), N0 = 0.04 cm−4 (2100–2200 UTC), and N0 = 0.02 cm−4 (2200–0000 UTC), respectively, for each hourly sequence.

Figure 9 displays retrieved and simulated vertical profiles of integral parameters of the DSD. For the reflectivity (Fig. 9a), the synthetic model vertical profiles are superposed by all the retrieved vertical profiles. The model profiles overlap the retrieved profiles with exceptions of erroneous artifacts of the retrieved reflectivity profile below h = 1000 m. Similarly, results for the rain rate (Fig. 9b) show the same overlapping behavior with profiles centered on 2 (2000 UTC), 4 (2100 UTC), 2 (2200 UTC), and 1.75 mm h−1 (2300 UTC). The differences observed for the liquid water content (Fig. 9c) are due to the choice of an exponential DSD instead of a gamma DSD, and by the fact that the concentration of small drops is more important in the case of an exponential DSD. The liquid water content (i.e., equal to the first-order moment of the DSD) is more sensitive than the reflectivity (i.e., proportional to the second-order moment of the DSD) on the left-hand side of the DSD spectra. This sensitivity analysis emphasizes the fact that simplifications in the model’s description of the flow field (e.g., neglecting horizontal transport and the local influence of the turbulence) can explain differences observed between simulated and retrieved vertical profiles of the reflectivity, rain rate, and liquid water content. In addition, the assumed shape of the DSD (i.e., exponential for the sensitivity analysis versus normalized gamma for radar retrieved data) is found to play a significant role in the discrepancies observed between modeled and retrieved vertical profiles for the rain rate and the liquid water content.

b. Model comparison with ground-based rainfall data measured by the disdrometer

The results presented above show a good agreement between modeling results obtained with the 1D rain shaft model and estimates retrieved from the vertically pointing radar observations. However, the main limitation of vertically pointing radars is their lack of observations in the lower part of the atmospheric column. As a general rule, lower-frequency radars have larger low-altitude data gaps. Because of this lack of observations it is not possible to determine the value of the mean air velocity from the 50-MHz profiler, which handicaps the reliability of the retrieved DSD close to the ground. For this dataset, the retrieved DSDs at levels below h = 1500 m assume that the vertical air velocity is constant with altitude and equal to the air velocity measured at the lowest reliable gate. This assumption leads to some unrealistic results in the retrieval of integral parameters as can be seen in Fig. 6 for (h = 500 m).

One way in which the rain shaft model can aid VPR retrieval is by providing a physically based dynamical interpolator at low levels to infer the shape of the DSD and track the evolution of integral parameters. To evaluate this assertion, simulations were performed using a 1-km column. Boundary conditions are imposed at the top of the shaft (at h = 1 km) based on the DSDs retrieved from VPR at the same height. The choice of a 1-km column is because few measurement artifacts are observed at this level (h = 1 km) thus providing confidence in the retrieved shape of the DSDs. Next, the modeling results were compared at ground level with measurements performed using a JWD.

Figure 10 displays the integral parameters Z (Fig. 10a), RNT (Fig. 10b), and LWC (Fig. 10c) derived from the disdrometer observations and from model simulations at ground level (h = 0 m), that is at the bottom of the shaft. On average, the differences between the time series for the evolution of integral parameters (Z, RNT, and LWC) predicted by the model at ground level and observed by the disdrometer are on the order of 3% (dBZ scale) for the reflectivity and 20% for the rain rate and the liquid water content. Note that despite the application of postprocessing corrections (Tokay et al. 2003), the disdrometer is expected to undersample the actual DSD because it cannot detect very small raindrops. The model adequately captures the general trend of the temporal evolution of integral parameters (Z, RNT, and LWC), including the time shift between the time series of VPR retrieved (at h = 1000 m) and JWD observed (at h = 0 m) integral properties (Z, RNT, and LWC; not shown).

The residuals obtained for the integral properties (Z, RNT, and LWC) between modeled and JWD time series are smaller and of the same order of magnitude as those between modeled and VPR retrieved time series [residuals of about 5% for Z (dBZ, 20%–30% in linear scale), and about 20%–50%] for RNT and LWC between modeled and VPR retrieved time series. A like-minded comparison between VPR and JWD retrievals (not shown) indicates that the residuals are of the same order of magnitude as those reported for (model – VPR) and (model – JWD). These experimental VPR – JWD residuals are consistent with the VPR – JWD reflectivity time series analysis of Gage et al. (2004) indicating about 2.1- and 0.4-dBZ rms difference between 1-min JWD and VPR reflectivities, respectively, and thus provides a measure of the experimental uncertainties of the data presented in this work.

Table 2 summarizes the values of the correlation coefficient for the integral parameters (Z, RNT, and LWC) between model–VPR, model–JWD, and JWD–VPR time series at different levels throughout the 1-km shaft. A correlation coefficient equal to 1 between model and radar time series corresponds to the top of the rain shaft where boundary conditions are imposed. Despite the presence of artifacts, model and measured radar reflectivity time series exhibit a good degree of correlation. A similar behavior is observed between radar and disdrometer time series with maxima of 0.81, 0.76, and 0.72 obtained for Z, RNT, and LWC time series close to the top of the shaft. A comparison of model and disdrometer time series for Z, RNT, and LWC shows that the correlation coefficient increases with increasing fall distance (i.e., as we get closer to the ground) and tends to a value of about 0.84 for Z and 0.87 for RNT and LWC.

Model and disdrometer DSDs are compared at different times during the period 2000–0000 UTC in Fig. 11. For drop diameters greater than d = 0.1 cm, the shapes of modeled and JWD DSDs match closely for drop number concentrations between 5.10−5 and 10−4 cm−4. The major differences between model and disdrometer DSDs are observed in the left-hand side of the spectra for the small raindrops. The JW disdrometer detects drops impacting an aluminum surface of 50 cm2, and an electric pulse resulting from the displacement of the sensor surface supporting device is measured. The electric pulse, proportional to a value between the force of the drops impacting the sensor and the mechanical momentum, is sorted into 20 class sizes equally spaced in terms of drop diameter (Sheppard 1990; McFarquhar and List 1993). To account for the instrument dead time, a mathematical correction (Sheppard and Joe 1994; Sauvageot and Lacaux 1995) is applied to increase the number of drops detected in the smallest size categories. Nevertheless, only drops above a certain threshold size (typically 0.03 cm in diameter) can be detected by the disdrometer. Furthermore, the threshold diameter is rain-rate dependent and can be up to d = 0.1 cm in case of heavy rain rate (McFarquhar et al. 1996). For the period considered 2000–0000 UTC, the threshold detection size for raindrops varies between 0.04 and 0.05 cm for light-to-moderate rain rate (<7 mm h−1).

To help with assessing the impact of the disdrometer measurement threshold for small drop diameters, Fig. 12 displays the time evolution of model and disdrometer drop number concentrations (M0). When computed over all the 40 bins (covering a diameter range from d = 0.01 to 0.7 cm), the temporal average of the difference between modeled and disdrometer integral parameters is on the order of 45% for M0, 4% for LWC, 6% for RNT, and 2% for Z for the period 2000–2400 UTC. The difference M0 varies widely in time from 45%, 90%, 27%, and 25% for hours 2000, 2100, 2200, and 2300 UTC, respectively. The larger errors occur at the beginning of the rain event when the concentration of small drops is larger and the errors decrease with time along with the concentration of small drops. If the model–JWD comparison is undertaken using exclusively the model bins that correspond to disdrometer channels that detect a drop, the difference in M0 is greatly reduced to an average value of 15% for the 2000–0000 UTC rain event with hourly averages of 5%, 27%, 5%, and 1%. The results show that there is an improvement in the order of 90% in the estimation of rain rate and liquid water content using the model in the lower atmosphere h ≥ 500 m (Table 2).

This exercise shows that indeed there is great potential value in using a rain shaft model with detailed microphysics to complement the standard radar retrieval algorithms where radar retrieval is ambiguous, or excessively plagued by uncertainty. Furthermore, there is clearly great need to improve the measurement sensitivity of disdrometers for light rainfall. This simulation also provided insight into the possible consequences of using disdrometer data to calibrate or constrain radar retrieval algorithms, which can result in the introduction of inconsistent physics due to the lack of sensitivity of the disdrometer to measure accurately small drop population. Given the lack of ability to describe the dynamics of the very small raindrop populations in current parameterizations (Barros et al. 2008), this is an area where there is great need for fundamental experimental research both in terms of measurement technology as in terms of conceptual understanding.

5. Conclusions and perspectives

In this paper, results from a homogeneous rain shaft model were presented and compared with field experiment radar rainfall data collected at Darwin, Australia, on 23 January 2006 during TWP-ICE for rain events passing through the observation site from 1700 to 0000 UTC. Available rain events included convective rain sequences with the presence of important updrafts and stratiform rain sequences with almost free-fall conditions. The rain event was characterized by a wide range of intensity from light (2–5 mm h−1), to intermediate (10–15 mm h−1) and heavy rainfall (20–50 mm h−1). This study focused mainly on a stratiform rain sequence from 2000 to 0000 UTC with weak vertical velocities (updrafts–downdrafts).

The comparison between modeling results and VPR retrieved data shows generally good agreement for the temporal evolution for vertical profiles of integral quantities (Z, RNT, and LWC) at fixed heights throughout the 1D rain shaft, although differences increase with increasing distance from the h = 4-km level at which retrieved DSDs were imposed as top boundary conditions. Model-retrieved data residuals were found to be around 5% (dBZ scale, or 20%–25% linear scale) for the reflectivity (Z), and most of the time fall within the range of uncertainty expected with radar measurements and with the data retrieval procedure (±1 dBZ). For other integral parameters, differences were comparable to estimated experimental uncertainties (20%–25% for the rain rate and the liquid water content). Results from sensitivity analysis using generated random DSDs at the top of the rain shaft model based on retrieved DSD parameters indicate that the variability of the simulated integral parameters were consistent with the variability of the retrieved integral parameters. Retrieved data present uncertainties because of instrument accuracy, resolution and assumptions, and specific formulation of the retrieval algorithm used at this time.

Overall, the conclusion is that, despite large simplifications in the representation of the wind fields, the model is capable of reproducing the dynamic temporal and spatial evolution of VPR-based retrievals of reflectivity and other integral parameters (LWC and RNT) using realistic boundary conditions imposed at the top of the rain shaft. The results presented in this study suggest that the rain shaft model with explicit microphysics could be used to describe warm rain microphysical processes in conjunction with radar rainfall retrieval algorithms. This study presented a proof-of-concept application of the rain shaft model as a physically based interpolator to improve radar retrieval at low levels (i.e., between the cloud base and the land surface). Other integral parameters of the DSD such as the rain rate and the liquid water content can be obtained from the rain shaft model as well as the DSD spectra without any assumption of the expected shape of the DSD (e.g., Gaussian or exponential). Future developments to the rain shaft model include the extension to a three-dimensional heterogeneous domain in order to account for the influence of horizontal wind and wind shear effects in the case of convective rainfall. For instance one of the next steps could consist in combining a detailed transport model that will account for the transport of drops in the atmosphere with the current microphysical model that will account for the impact of the spatial heterogeneity of the flow field on the dynamic of microphysical processes.

Finally, particularly useful insight was gained from this research with regard to the use of JWD observations by confirming previous concerns with regard to the implications of lack of measurement capacity at small drop sizes, and how this can affect the use of JWD data directly to calibrate or constrain radar algorithms. We suggest that because it is difficult to determine a priori the critical fall distance below which the radar looses sensitivity, and because of aforementioned caveats regarding the use of disdrometers, a model such as the one used here provides an alternative to standard algorithms to derive surface rainfall estimates from radar reflectivity estimates at higher levels alone. Furthermore, a 3D configuration of the model should be suitable to estimate rainfall intensity over a large horizontal area using satellite reflectivity [i.e., Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR)] as an input for the model.

Acknowledgments

This research was supported in part by NASA Grant NNX07AK40G with the second author. The third author was supported in part by NASA Precipitation Measurement Mission (PMM) Grant NNX07AN32G and in part by NOAA’s contribution toward the NASA PMM program. The Darwin 50-MHz profiler is owned and operated by the Australian Bureau of Meteorology (BOM). The Darwin 920-MHz profiler is owned by NOAA and is maintained and operated by BOM. The profiler and disdrometer observations collected during TWP-ICE were supported by the NASA Tropical Rainfall Measuring Mission (TRMM) and BOM. The authors express their gratitude to Dr. V. Chandrasekar from Colorado State University for valuable and insightful suggestions in the latest stage of this work and to Dr. Greg McFarquhar from the University of Illinois at Urbana–Champaign and an anonymous reviewer for their valuable and constructive comments.

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Fig. 1.
Fig. 1.

Time–height cross section of (a) reflectivity, (b) mean Doppler velocity, and (c) spectrum variance from 920-MHz profiler for 1000 UTC 23 Jan–0000 24 Jan 2006.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 2.
Fig. 2.

Doppler velocity spectra from (a) 50- and (b) 920-MHz profiler. Air motion is estimated from 50-MHz profiler and plotted as asterisks on 920-MHz profiler spectra. DSD is estimated from 920-MHz profiler spectra that observe Doppler motion of hydrometeors.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 3.
Fig. 3.

Evolution of retrieved integral parameters below the bright band (at h = 4000 m) for the rain event observed from 2000 UTC 23 Jan to 0000 24 Jan 2006 in Darwin, Australia: (a) Z, (b) RNT, (c) LWC, and (d) M0.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 4.
Fig. 4.

Evolution of the retrieved parameters of the modified Gamma DSD below the bright band (at h = 4000 m). (bottom) The median drop diameter (d0) and the shape factor (μ). (top)The corresponding DSDs grouped by (left to right) hours of the rain event: 2000–2100, 2100–2200, 2200–2300, and 2300–0000 UTC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 5.
Fig. 5.

Comparison between modeled (M) and retrieved (R) vertical parameters for (left to right) 2213, 2233, and 2253 UTC for (a) Z, (b) RNT, and (c) LWC. The residual: Res. = (MR)/R is displayed in the last column. Notation: 22.XX = 22XX UTC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 6.
Fig. 6.

Evolution of (left) Z, (middle) RNT, and (right) LWC for (top to bottom) different fall distances (Dfall)—from 100 to 3500 m—from the top of the rain shaft [i.e., height (h) from ground level] for 2000–0000 UTC: model results and retrieved data.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 7.
Fig. 7.

Scatterplots for (left) Z, (middle) RNT, and (right) LWC for model results and radar data for different fall distances (Dfall)—from 100 to 2500 m—from the top of the rain shaft [i.e., height (h) from ground level] for 2000–0000 UTC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 8.
Fig. 8.

Values of the slope of the DSD parameter Λ for randomly generated exponential DSDs for the period 2000–0000 UTC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 9.
Fig. 9.

Comparison of vertical profiles from model (randomly generated MP DSD) and retrieved (TWP-ICE) at the top of the rain shaft (h = 4000 m) for the period 2000–0000 UTC for (a) Z, (b) RNT, and (c) LWC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 10.
Fig. 10.

Comparison of the evolution of integral parameters retrieved from the JWD disdrometer at ground level) with model results); percent difference between model and JWD: (a) Z, (b) RNT, and (c) LWC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 11.
Fig. 11.

Comparison of DSD at ground level between model (times signs) and JWD (plus signs) at (a) 2020, (b) 2040, (c) 2140, (d) 2120, (e) 2220, (f) 2240, (g) 2320, and (h) 2340 UTC.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Fig. 12.
Fig. 12.

Comparison of M0 for 2000–0000 UTC between model and JWD. For modeling results, integral parameters are computed using a summation on all 40 bins and a summation on bins corresponding to JWD channels for which drops are detected.

Citation: Journal of Applied Meteorology and Climatology 47, 11; 10.1175/2008JAMC1801.1

Table 1.

Correlation coefficient for Z, RNT, and LWC time series between modeling results and vertically pointing radar at various fall distance (Dfall) from cloud top (h = 4000 m).

Table 1.
Table 2.

Correlation coefficient for Z, RNT, and LWC time series between modeling results and vertically pointing radar, modeling results and disdrometer, and vertically pointing radar and disdrometer at various fall distance (Dfall) from the top of the shaft (h = 1000 m); NA indicates not available.

Table 2.
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