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  • View in gallery

    An example of an SMC system observed from ground-based remote sensing measurements over the ACRF NSA Barrow site on 5 Nov 2007. (a) MPL backscatter, (b) MPL lidar depolarization ratio (DEP), (c) MMCR Ze, and (d) MMCR VD profiles. Positive Doppler velocity is downward. Black lines indicate the radar-detected cloud tops and lidar-detected liquid-dominated mixed-phase-layer bases.

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    Sounding (temperature, equivalent potential temperature, and relative humidity with respect to liquid) measured at 0506 UTC 5 Nov 2007 over the ACRF NSA Barrow site.

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    (a) A conceptual model of ice crystal growth along their fallout trajectory in SMCs. (b) A snapshot of the MPL backscatter, MMCR Ze, MMCR VD, and relative humidity with respect to the ice profiles shown in Fig. 1.

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    The schematic diagram of the 1D ice growth model for SMCs.

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    The ice crystal mass after 3, 5, 10, 15, 20, and 25 min of growth at temperatures ranging from −2° to −30°C. The initial radius of the ice crystal is 4 μm. The wind tunnel measurements from Takahashi et al. (1991) are plotted with different signs for comparison. The spherical ice growth at 25 min is also shown (dashed line).

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    As in Fig. 5, but for the (a) a-axis and (b) c-axis length growth with time.

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    Mean LWP values (asterisks) of SMCs as a function of CTT based on 4-yr ground-based microwave radiometer measurements at the ACRF NSA Barrow site. The boxes represent 25%, 50%, and 75% of the LWP at each CTT.

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    Comparisons of reflectivity-weighted Vt of ice crystals from 1D ice growth model simulations (black solid lines) with 4-yr statistics of MMCR-measured Doppler velocities at each CTT. The red dashed lines indicate mean values. The red boxes represent 25%, 50%, and 75% of the MMCR Doppler velocities at each distance below the cloud top.

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    As in Fig. 8, but for the Ze_n profile comparisons. The 1D ice growth model simulations assuming spherical particles (black dashed lines) are also plotted.

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    The Ze_layer values calculated from the 1D ice growth model assuming Ni of 1 L−1 (black line) and from 4-yr MMCR measurements over the ACRF NSA Barrow site (red line) at each CTT. The 1D ice growth model simulations without riming growth (blue line) are also plotted to show the relative contributions of diffusional growth and riming growth.

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    Sensitivities of the simulated Ze_layer to parameterizations in the 1D ice growth model: (a) sensitivity to ±20% uncertainties in ice growth rate, (b) sensitivity to ±25% uncertainties in Vt parameterization, (c) sensitivity to and σ, and (d) sensitivity to ±60% LWP variations.

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    An SMC system detected during the CAMPS field campaign on 17 Feb 2011. (a) The Ze structure from WCR. The white line near 0.7 km above flight level is the radar-detected cloud top. (b) WCL backscattering power. (c) The 2D-C-measured Ni at flight level (black), WCR-measured Ze_layer (blue), and retrieved Ni (red).

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    Comparisons of the retrieved Ni with 2D-C measurements for the three SMC systems during ICE-L (black), ISDAC (red), and CAMPS (green) field campaigns. The legend on the top left indicates the field campaign name, date, and mean CTT. The dashed lines are the factor-of-2 lines.

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    Comparison of the retrieved Ni from the three-dimensional CRM-simulated Ze_layer with Ni in the model based on an M-PACE case. (a) The Ze profiles from the radar simulator in the three-dimensional CRM, (b) Ni profiles from the three-dimensional CRM simulation, (c) time series of the mean Ni within 500 m of the cloud top from the three-dimensional CRM simulations and Ze_layer retrievals, and (d) comparison of Ni between the three-dimensional CRM simulations and Ze_layer retrievals. The black dashed lines are the factor-of-2 lines.

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Ice Concentration Retrieval in Stratiform Mixed-Phase Clouds Using Cloud Radar Reflectivity Measurements and 1D Ice Growth Model Simulations

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  • 1 Department of Atmospheric Science, University of Wyoming, Laramie, Wyoming
  • 2 National Center for Atmospheric Research,* Boulder, Colorado
  • 3 Pacific Northwest National Laboratory, Richland, Washington
  • 4 Department of Atmospheric Science, University of Wyoming, Laramie, Wyoming
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Abstract

Measurements of ice number concentration in clouds are important but still pose problems. The pattern of ice development in stratiform mixed-phase clouds (SMCs) offers an opportunity to use cloud radar reflectivity (Ze) measurements and other cloud properties to retrieve ice number concentrations. To quantify the strong temperature dependencies of ice crystal habits and growth rates, a one-dimensional (1D) ice growth model has been developed to calculate ice diffusional growth and riming growth along ice particle fallout trajectories in SMCs. The radar reflectivity and fallout velocity profiles of ice crystals calculated from the 1D ice growth model are evaluated with the Atmospheric Radiation Measurements (ARM) Climate Research Facility (ACRF) ground-based high-vertical-resolution radar measurements. A method has been developed to retrieve ice number concentrations in SMCs at a specific cloud-top temperature (CTT) and liquid water path (LWP) by combining Ze measurements and 1D ice growth model simulations. The retrieved ice number concentrations in SMCs are evaluated using integrated airborne in situ and remote sensing measurements and three-dimensional cloud-resolving model simulations with a bin microphysical scheme. The statistical evaluations show that the retrieved ice number concentrations in the SMCs are within an uncertainty of a factor of 2.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Damao Zhang, Department of Atmospheric Science, University of Wyoming, 1000 E. University Ave., Laramie, WY 82071-2000. E-mail: dzhang4@uwyo.edu

Abstract

Measurements of ice number concentration in clouds are important but still pose problems. The pattern of ice development in stratiform mixed-phase clouds (SMCs) offers an opportunity to use cloud radar reflectivity (Ze) measurements and other cloud properties to retrieve ice number concentrations. To quantify the strong temperature dependencies of ice crystal habits and growth rates, a one-dimensional (1D) ice growth model has been developed to calculate ice diffusional growth and riming growth along ice particle fallout trajectories in SMCs. The radar reflectivity and fallout velocity profiles of ice crystals calculated from the 1D ice growth model are evaluated with the Atmospheric Radiation Measurements (ARM) Climate Research Facility (ACRF) ground-based high-vertical-resolution radar measurements. A method has been developed to retrieve ice number concentrations in SMCs at a specific cloud-top temperature (CTT) and liquid water path (LWP) by combining Ze measurements and 1D ice growth model simulations. The retrieved ice number concentrations in SMCs are evaluated using integrated airborne in situ and remote sensing measurements and three-dimensional cloud-resolving model simulations with a bin microphysical scheme. The statistical evaluations show that the retrieved ice number concentrations in the SMCs are within an uncertainty of a factor of 2.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Damao Zhang, Department of Atmospheric Science, University of Wyoming, 1000 E. University Ave., Laramie, WY 82071-2000. E-mail: dzhang4@uwyo.edu

1. Introduction

The production of ice crystals in clouds has significant implications for their precipitation efficiency, lifetimes, and radiative properties (Lohmann and Feichter 2005; Gettelman et al. 2012). At temperatures between −40° and 0°C, liquid droplets and ice particles can coexist in clouds by forming mixed-phase clouds. Based on space-borne and ground-based remote sensing measurements, previous studies have revealed that mixed-phase clouds have large global coverage (Wang 2013) and are the dominant cloud type over polar regions (Curry et al. 1996; Zhao and Wang 2010). Shupe et al. (2006) showed that mixed-phase clouds occur approximately 40% of the time in the western Arctic by using ground-based remote sensing measurements. The high frequency of occurrence of mixed-phase clouds suggests that they play an important role in Earth’s climate system. However, mixed-phase clouds are still poorly represented in climate models (Klein et al. 2009; Morrison et al. 2009). Because of the lower saturation vapor pressure over ice than over liquid, ice particles grow at the expense of liquid droplets in the mixed-phase layer, which is known as the Wegener–Bergeron–Findeisen (WBF) process. With high-enough ice number concentrations Ni, the WBF process can sweep out or evaporate all of the liquid droplets and cause rapid glaciation of the mixed-phase clouds (Korolev and Isaac 2003; Morrison et al. 2011). Previous cloud-resolving model (CRM) simulations have shown that the glaciation temperature of mixed-phase clouds is highly sensitive to Ni and ice crystal habit (Korolev and Isaac 2003; Sulia and Harrington 2011). Global climate model (GCM) simulations show that the variation of the glaciation temperatures of clouds from 0° to −40°C yields approximately 4 and 8 W m−2 differences in longwave and shortwave cloud radiative forcing, respectively (Fowler et al. 1996; Gettelman et al. 2012). In addition, the intercomparison of model simulations of stratiform mixed-phase clouds suggests that the common underpredictions of liquid water content are related to the simplified treatments of Ni (Klein et al. 2009; DeMott et al. 2010; Fridlind et al. 2012). Improved Ni parameterizations in models can lead to more realistic simulations of the liquid and ice water distributions in clouds and of radiative forcing (Klein et al. 2009; DeMott et al. 2010). Therefore, accurate representations of Ni are critical for improving mixed-phase cloud simulations in models.

The primary heterogeneous ice nucleation processes are still not well identified or reliably parameterized in models because of the complicated pathways of ice formation and properties of ice nuclei (IN) (Gregory and Morris 1996; Cantrell and Heymsfield 2005; Hoose and Möhler 2012; Murray et al. 2012). Based on 14 years of aircraft in situ continuous-flow diffusion chamber (CFDC) IN measurements, DeMott et al. (2010) developed a new IN parameterization that is a function of both the cloud temperature and coarse aerosol (diameters larger than 0.5 μm) number concentrations. This parameterization reduces the variation of the IN concentration at a given temperature from a factor of approximately 103 to less than 10, and the remaining variability is considered to be caused by variations in the aerosol chemical composition or other factors. However, the relationship between IN and Ni must be further investigated (Baumgardner et al. 2012). The intercomparison of 17 single-column model simulations and 9 cloud-resolving model simulations of a mixed-phase stratocumulus cloud system shows that there are differences of approximately five orders of magnitude in the predicted Ni among the different models (Klein et al. 2009); these differences are directly linked to a greater-than-one-order-of-magnitude difference among the different Ni parameterizations that are used by different models. Therefore, improving the understanding of ice nucleation processes in clouds and the constraining parameterizations of Ni in models is urgently required.

The measurements of Ni in clouds are primarily generated from airborne in situ optical particle size spectrometers, such as the two-dimensional cloud (2D-C) and precipitation (2D-P) probes and the cloud imaging probe (CIP) (Baumgardner et al. 2011). These in situ probes provide relatively accurate measurements of Ni in clouds that are critical for the understanding of ice nucleation processes and improving the Ni parameterization in models (Cantrell and Heymsfield 2005). However, ice crystal shattering on the inlets and arms of the particle probes (Jensen et al. 2009) and aircraft-produced ice particles (APIP) (Heymsfield et al. 2011b) lead to overestimates of ice concentrations in clouds by airborne optical spectrometer measurements. Previous studies have suggested that Ni can be overestimated by two orders of magnitude as a result of the shattering effect (Korolev et al. 2011, 2013). Recently, the improved design of the tips of particle probes has been shown to effectively reduce the artifact generation. Combining the modified tips with particle interarrival-time correction algorithms has provided the best known method to mitigate ice-shattering effects (Korolev et al. 2011). A reanalysis of the historical Ni measurements has been proposed wherein the shattering effects would be treated carefully (Korolev et al. 2013). In addition, in situ probes generally have small sample volumes, which might lead to significant underestimates in the measured Ni, especially when Ni is low (Baumgardner et al. 2011). Furthermore, in situ aircraft measurements only cover short sampling periods and in limited regions, which makes it difficult to accumulate a large database for the statistical study of Ni in clouds and determine ice generation processes under various conditions, such as different dynamical environments and aerosol loadings.

Considering the large variations in Ni parameterizations at specific temperatures and the limitations and potential errors of measuring Ni from airborne in situ measurements, reasonably accurate estimations of Ni can be achieved using remote sensing observations, despite the inherent uncertainties associated with remote sensing measurements. The main advantage of estimating Ni with remote sensing measurements is their large spatial and temporal coverage. Nevertheless, a critical step for reliably estimating Ni from remote sensing measurements is to select the “right clouds” for analysis. Stratiform mixed-phase clouds (SMCs) represent a relatively simple scenario for analyzing the ice production characteristics in clouds and for retrieving Ni because of their less-complex dynamic environments and well-defined vertical thermodynamic structures (Heymsfield et al. 1991, 2011a; Fleishauer et al. 2002). During the Ice in Clouds Experiment–Layer Clouds (ICE-L) field campaign, the SMCs are regarded as an ideal target for studying the primary ice formation mechanism in clouds (Heymsfield et al. 2006). Over polar regions, extensive and persistent SMCs are common (de Boer et al. 2011; Morrison et al. 2012). Over the tropics and midlatitude regions, midlevel SMCs also occur approximately 4.5% of the time, based on satellite remote sensing measurements (Zhang et al. 2010). Recently, observational studies of ice nucleation, cloud evolution processes, and aerosol (focus on dust) impacts on ice nucleation in SMCs were conducted extensively over many regions (Fleishauer et al. 2002; Ansmann et al. 2009; Zhang et al. 2010, 2012; Bühl et al. 2013).

In this study, we develop an algorithm to retrieve Ni in SMCs by combining radar reflectivity (Ze) measurements with a one-dimensional (1D) ice growth model. The paper is organized as follows. Section 2 describes a conceptual model of the ice growth along ice particle fallout trajectories in SMCs. Section 3 describes the development of a 1D ice growth model for calculating the ice diffusional growth and riming growth along fallout trajectories and the corresponding Ze structure. The combination of the modeled and measured Ze provides the Ni estimations in the SMCs. Validations of our 1D ice growth model simulations with laboratory measurements and ground-based remote sensing observations are provided, and the sensitivity of the modeled Ze structure to model assumptions and cloud properties are also discussed. Section 4 provides evaluations of the retrieved Ni in the SMCs using in situ measurements and simulations of a three-dimensional cloud-resolving model with a bin microphysical scheme. Finally, a discussion and summary of the results is presented in section 5.

2. Ice growth along particle fallout trajectories in SMCs

Previous long-term ground-based remote sensing observations and in situ measurements have shown that there is usually a liquid-dominated layer at the top of an SMC, and the depth of the liquid-dominated layer is generally less than 500 m (Heymsfield et al. 1991; Fleishauer et al. 2002; Wang et al. 2004). Partly as a result of the relatively colder temperatures, ice crystals primarily form from the liquid phase at the top of the SMCs (Carey et al. 2008; Fan et al. 2011; Westbrook and Illingworth 2011) and grow rapidly in a liquid-saturated and ice-supersaturated environment; the crystals then fall out of the liquid-dominated mixed-phase layer (Fleishauer et al. 2002; Wang et al. 2004; Noh et al. 2013). Below the liquid-dominated mixed-phase layer, ice crystals continue to grow and fall until they reach the level below the ice saturation condition and then start to sublimate. (Figure 2 shows an example of a sounding profile for an Arctic SMC system that supports the SMC vertical structure described above.)

Although SMC systems are difficult to identify with conventional observations, they can be detected reliably by combining lidar and radar measurements (Wang et al. 2004; Ansmann et al. 2009; Zhang et al. 2010). Because it uses different wavelengths, lidar is more sensitive to the liquid phase (high number concentration of small-sized particles), whereas radar is more sensitive to large-sized ice particles, even when there is a much lower number concentration of ice particles than liquid droplets. Therefore, combining lidar and radar measurements provides the vertical structure of the cloud phase as well as the ice evolution in the SMCs. Hogan et al. (2003) employed 18 months of ground-based lidar and radar measurements to study the characteristics of SMCs over two midlatitude locations. Taking advantage of the collocated space-borne Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) lidar and CloudSat radar observations, Zhang et al. (2010) presented a global climatology of midlevel SMCs and the hydrometeor phase partition.

The Atmospheric Radiation Measurement (ARM) Climate Research Facility (ACRF) provides state-of-the-art measurements of the macrophysical and microphysical properties of aerosol and clouds and their radiation forcing (Mather and Voyles 2013). The North Slope of Alaska (NSA) Barrow facility (71.34°N, 156.68°W) was built in 1997, and it provides long-term observations of SMCs. In our study, ground-based micropulse lidar (MPL) and millimeter-wavelength cloud radar (MMCR) measurements over the NSA Barrow site are analyzed to identify SMCs and provide Ze measurements for the validation of simulated Ze structure, which will be discussed in section 4. The characteristics of the instruments at the ACRF NSA Barrow site and their measurements are listed in Table 1. Data for the period between October 2006 and July 2010 are analyzed.

Table 1.

Instruments at the ACRF NSA Barrow site used in this study.

Table 1.

Figure 1 shows an example of an SMC system observed at the NSA site on 5 November 2007. A liquid-dominated layer at the top of the SMC is identified using the lidar backscattering profile “slope method” (Wang and Sassen 2001). Radar reflectivity thresholds are used to detect ice particles below the liquid-dominated layer (Zhang et al. 2010). A liquid-dominated layer is observed at altitudes between approximately 0.7 and 1.0 km AGL, with tops varying from approximately 0.9 to 1.2 km AGL based on strong MPL backscattering (Fig. 1a) and low lidar depolarization ratios (Fig. 1b). However, the existence of radar signals above 1 km suggests that these clouds have tops of approximately 1.2 km AGL with little variability. Lidar signals are fully attenuated most of the time, and conspicuous ice particles below the liquid-dominated layer are detectable from lidar depolarization (Fig. 1b) and radar Ze measurements (Fig. 1c). The increase of both Ze and Doppler velocity with distance below the cloud top in Figs. 1c and d reflects the growth of ice crystals along the fall trajectory. Sounding measurements at 0506 UTC 5 November 2007 over the ACRF NSA Barrow site are shown in Fig. 2. A temperature inversion of approximately 4 K is observed at the top of the SMCs (~1.2 km AGL), with a dramatic decrease of relative humidity above the inversion layer. The top boundary layer of the SMCs is well mixed, which is indicated by the constant equivalent potential temperature. The surface and cloud-top temperatures (CTT) of the SMCs are about −11° and −19°C, respectively. The liquid-saturated layer extends approximately from 1.2 down to 0.6 km, which is consistent with the MMCR and MPL measurements. This SMC system persists for more than 14 h.

Fig. 1.
Fig. 1.

An example of an SMC system observed from ground-based remote sensing measurements over the ACRF NSA Barrow site on 5 Nov 2007. (a) MPL backscatter, (b) MPL lidar depolarization ratio (DEP), (c) MMCR Ze, and (d) MMCR VD profiles. Positive Doppler velocity is downward. Black lines indicate the radar-detected cloud tops and lidar-detected liquid-dominated mixed-phase-layer bases.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

Fig. 2.
Fig. 2.

Sounding (temperature, equivalent potential temperature, and relative humidity with respect to liquid) measured at 0506 UTC 5 Nov 2007 over the ACRF NSA Barrow site.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

A conceptual model of ice crystal growth along the fall trajectory in SMCs is presented in Fig. 3a. The model is based on previous in situ measurements and ground-based remote sensing observations. Figure 3b shows a snapshot of the MPL backscattering power, MMCR Ze, MMCR Doppler velocity VD, and relative humidity with respect to ice (RHice) profiles of the SMC shown in Fig. 1. The RHice profile is calculated using the closest sounding data, and the HSi and HSw correspond to the ice and water saturation levels, respectively. From Fig. 3b, Ze increases gradually approximately from −30 dBZ at the cloud top (~1.2 km AGL) to −8 dBZ at the liquid-dominated layer base (~0.6 km AGL). As expected, below the liquid-dominated layer, Ze continues to increase to approximately −2 dBZ at a height of about 0.4 km AGL, which is slightly higher than the ice saturation level, and then starts to decrease gradually. Similarly, the MMCR VD increases approximately from 0 m s−1 at the cloud top to 1.1 m s−1 at the ice saturation level, where it remains almost constant as ice particles continue to fall. The Ze and VD profiles provide solid support for ice growth along the fallout trajectories described in the conceptual model.

Fig. 3.
Fig. 3.

(a) A conceptual model of ice crystal growth along their fallout trajectory in SMCs. (b) A snapshot of the MPL backscatter, MMCR Ze, MMCR VD, and relative humidity with respect to the ice profiles shown in Fig. 1.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

3. 1D ice growth model for SMCs and evaluations

Once an ice crystal is formed in an SMC, it starts to grow through water vapor diffusion. Under liquid-saturated environments and at the same CTT, the ice crystal growth rates are expected to be similar because of the small vertical air motions and weak turbulence in SMCs (Heymsfield et al. 1991; Fleishauer et al. 2002). This assumption has been evaluated with integrated airborne remote sensing and in situ measurements. Zhang et al. (2012) showed that assuming the same ice-size distribution shape for similar SMCs in terms of CTT and liquid water path (LWP) only causes differences in the calculated Ze of 2 dBZ or less, compared with the values calculated using the 2D-C probe–measured ice crystal size distributions. Therefore, if we can further model the ice crystal growth trajectory in SMCs, we will be able to estimate the total ice crystal number concentration from the Ze structure. Because of the contribution of liquid droplets to the Ze at the cloud top and a typical depth of about 500 m for the liquid-dominated mixed-phase layer, we therefore use the mean Ze (Ze_layer) between the cloud top and 500 m below to retrieve the Ni in SMCs. The 500-m depth of the Ze_layer is also the same as the pulse width of the CloudSat radar, which makes it convenient to apply the algorithm to CloudSat radar measurements to study the global Ni characteristics in the future. Here, a 1D ice growth model for calculating the ice diffusional and riming growth and Ze structure of the SMCs at a specific CTT and LWP is presented. In the 1D model, the temperature and RH profiles are determined by CTT and LWP assuming adiabatic liquid water and temperature profiles (Westbrook and Illingworth 2013). Figure 4 shows the schematic diagram of the 1D ice growth model. From the 1D ice growth model, the ice crystal size and shape profiles along the fallout trajectory are obtained at a specific initial ice crystal size and shape, CTT, LWP, and updraft. The initial ice crystal size and shape and updrafts are parameterized as described in section 3c. Thus, the Ze structure in terms of the distance from the cloud top is calculated from the simulated ice crystal size distribution profile. Finally, we calculate the Ze_layer from the Ze profiles at each CTT.

Fig. 4.
Fig. 4.

The schematic diagram of the 1D ice growth model for SMCs.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

a. Vapor diffusional growth of ice crystals

From the Maxwell–Mason equation (Mason 1953; Fukuta and Walter 1970), the vapor mass diffusional growth rate of a spherical particle is expressed as
e1
where r is the radius of the particle, Si is the saturation with respect to ice, Ld is the latent heat of deposition, K is the thermal conductivity of air, Rυ is the specific gas constant of water vapor, T is the environmental temperature, Dυ is the diffusivity of water vapor in the air, ρs is the saturated vapor density at T, and fυ is the ventilation factor.

Previous laboratory experiments and in situ measurements show that wide varieties of ice crystal shapes occur under different environmental conditions (e.g., temperature T and RH) and include relatively simple hexagonal plates and columns, dendrites, rosettes, and complicated polycrystalline shapes that form at cold temperatures (Bailey and Hallett 2004, 2009, 2012; Korolev et al. 1999). The ice crystal shapes or growth habits are primarily temperature dependent. The primary ice crystal habits are plates (T > −4.0°C), columns (−4° > T > −8°C), plates (−8.1° > T > −22.4°C), and columns (T < −22.4°C). Isometric ice crystal growth habits occur at transition temperatures of approximately −4°, −8°, and −22°C (Fukuta and Takahashi 1999; Bailey and Hallett 2009, 2012). Recently, Bailey and Hallett (2009) revealed that ice crystal habits at temperatures between −20° and −40°C are more platelike and rarely column-like, which is contrary to previous ice growth habit diagrams at temperatures colder than −22°C. These laboratory data and in situ measurements provide vital information for the parameterization of ice growth habits in numerical cloud models (Fukuta and Takahashi 1999).

To account for the nonspherical shapes of ice crystals, proxies for the shapes of oblate and prolate spheroids with two primary semiaxes of a and c are often used for platelike and column-like ice crystals, respectively (Chen and Lamb 1994, hereafter CL94). The radius r in Eq. (1) is then replaced with C, which is the electrostatic capacitance by analogy to the electrostatic field (Lamb and Verlinde 2011). This parameterization is called the “capacitance model,” and C is a function of ice crystal size and shape. Direct laboratory measurements of C are available, and ice crystal growth rates predicted from the capacitance model are compared with the laboratory measurements at temperatures warmer than −22°C (Westbrook et al. 2008; Westbrook and Heymsfield 2011). However, one drawback of the capacitance model is that the aspect ratio, defined as , does not change with ice crystal size (Harrington et al. 2013a).

CL94 developed a parameterization of ice crystal habits that allows for the evolution of the a axis and c axis of the ice crystals with time and also with ice crystal size. To do this, CL94 proposed a “mass distribution hypothesis” based on the diffusional growth theory:
e2
where and is the ice inherent growth ratio (IGR), which is primarily dependent on the temperature under liquid water saturation conditions (CL94); αc and αa are the mass deposition coefficients ranging from 0 to 1 along the c axis and a axis, respectively. From this hypothesis, ϕ evolves with time and contains information on the ice crystal shape evolution process. Based on the definition of ϕ, it also reflects the ratio of water vapor flux for each axis (Sulia and Harrington 2011). Note that ϕ has a positive feedback on the evolution of ice crystal shapes. Physically, Eq. (2) indicates that the mass distributed on each axis is determined by the deposition coefficients and water vapor flux for that axis (Sulia and Harrington 2011). This is the primary hypothesis in CL94’s parameterization. Because CL94’s parameterization is able to describe the growth of the c axis and a axis, it can be used to simulate the growth habit of most ice crystals; therefore, it is physically more reliable. Recently, CL94’s parameterization has been incorporated into many cloud-resolving models to investigate the impacts of ice growth habits on the microphysical properties, phase partitions, and lifetimes of mixed-phase clouds (Avramov and Harrington 2010; Ervens et al. 2011; Sulia and Harrington 2011; Harrington et al. 2013b).

The best-fit values of the IGR from both laboratory measurements and in situ observations were developed by CL94 within a temperature range between −30° and 0°C. However, CL94 presented IGR values greater than 1, which indicated a column-like growth habit at temperatures colder than −22°C and is inconsistent with the observations shown in Bailey and Hallett (2009). Hashino and Tripoli (2008, hereafter HT08) derived IGR values by tuning them numerically to match the laboratory-measured ϕ over different growth times (Bailey and Hallett 2004) and extended the IGR data down to −40°C. They derived IGR values for both platelike and column-like polycrystalline shapes when the temperatures are colder than −20°C. IGR values greater than 1 at the temperature range between −4° and −8°C correspond to column-like ice crystal growth, whereas IGR values smaller than 1 at the temperature range between −8° and −20°C correspond to platelike growth. In general, as IGR values deviate from 1, more irregular shapes of ice crystals evolve and the ice crystal grows more rapidly (CL94; Sulia and Harrington 2011; Avramov and Harrington 2010). Comparing the IGR values from HT08 and CL94, the IGR values from HT08 are greater for platelike growth and smaller for column-like growth, which produces slower ice diffusional growth compared to that from CL94 at each CTT (HT08). In this study, we use IGR values from CL94 at temperatures warmer than −22°C and IGR values for platelike polycrystalline from HT08 at temperatures colder than −22°C.

Using Eq. (1) to calculate the ice diffusional growth, the ice crystal mass–dimension relationship is also required. CL94 presented a modified simple parameterization of the deposition density ρdep to connect the change of mass with the change of volume [Eq. (B5) in appendix B] based on previous laboratory measurements (Fukuta 1969). The parameterized ρdep depends on both temperature and ice supersaturation, and the ρdep parameterization from CL94 is used in this study. However, previous ice growth simulations have shown that the fast ice growth temperature range centered around−15°C was much broader than that from laboratory measurements (Sulia and Harrington 2011; Harrington et al. 2013b), which might be related to the ice density parameterization (i.e., ρdep) within this temperature range. It was suggested that improved ρdep parameterization is still required in the future (Harrington et al. 2013b).

The equations for iteratively calculating ice crystal growth rates are described in appendix B. Given the initial ice crystal size and shape and thermodynamic environments (e.g., temperature and RH profiles), the ice crystal mass and size along the growth time scale are obtained. For simplicity, only the ice crystal diffusional and riming growths are simulated. The aggregation processes and secondary ice production are not considered in the present work.

When an ice crystal grows large enough, it falls at its terminal velocity, and the ventilation effects become nonnegligible and must be considered. The ventilation factor fυ is expressed as (Hall and Pruppacher 1976)
e3
where is the Schmidt number and equals 0.632 for ice crystals, Re is the Reynolds number (see appendix C), is the kinematic viscosity, and η and ρair are the dynamic viscosity and density of air, respectively. In general, the ventilation effect increases the ice crystal diffusional growth rate.
Although the ice diffusional growth is the dominant ice mass growth mechanism, ice particles may grow through riming as they grow and fall through the liquid-dominated layer in SMCs. In laboratory cloud-tunnel experiments, riming growth has been observed to occur within 30 min of growth, especially at temperatures of approximately −10°C (Fukuta and Takahashi 1999). In this study, the riming growth of ice particles is parameterized following Heymsfield (1982). The riming growth rate is
e4
where A is the particle cross-sectional area normal to the airflow; Vt is the ice particle terminal velocity (see appendix C); E is the collection efficiency of a droplet d by the ice particle D; and LWC is the liquid water content, which is calculated from the LWP and CTT by assuming adiabatic clouds. The total ice mass growth rate is the sum of the diffusional growth rate and riming growth rate.

b. Parameterizations of ice crystal terminal velocity

Although ice diffusional growth rates have been calculated and compared with laboratory measurements (Fukuta and Takahashi 1999; Westbrook and Heymsfield 2011), studies of ice growth along fallout trajectory are still rare despite having significant applications in the remote sensing of ice cloud microphysical properties (Heymsfield et al. 2011a). To compare ice growth with the observed Ze, we must first calculate the ice crystal mass growth along its fallout trajectory instead of along the growth time scale, which requires the parameterization of the ice crystal Vt, vertical air velocity, and turbulence. Models generally use the Vt–dimension relationship in the form of
e5
where the coefficients α and β for a variety of ice crystal shapes have been measured in the laboratory in previous studies (Redder and Fukuta 1991). However, there are still large variations of Vt as a result of shape variations, even for the same ice particle dimension L. Mitchell (1996) proposed to relate Vt with L, mass m, and projected area normal to the fall direction A and developed coefficients for different size ranges and ice crystal shapes. So far, Mitchell’s parameterization has been widely used in cloud and climate models (Mitchell et al. 2011; Sulia and Harrington 2011). Heymsfield and Westbrook (2010, hereafter HW10) suggested that Mitchell’s parameterization overestimated the Vt for open-geometry ice particles. They made a simple modification of Mitchell’s parameterization to improve the Vt parameterization and showed that the modified parameterization predicts Vt with errors of less than 25% compared to laboratory measurements. Therefore, HW10’s parameterization of Vt is used to calculate the Ze vertical profile along the fallout trajectory in this study. The equations for calculating the ice crystal Vt are given in appendix C.

c. Evaluation of 1D ice growth model simulations

To evaluate the ice crystal diffusional growth parameterization along with growth time, we compare the simulated ice crystal mass growth with laboratory measurements using a wind-tunnel cloud chamber from Takahashi et al. (1991) under different temperatures, which is shown in Fig. 5. The laboratory experiments were performed under pressures of approximately 1000 hPa, and the cloud droplets in the chamber were approximately 4 μm in radius on average. The model calculations of mass as a function of time are performed with the same parameters as in the laboratory experiments at each fixed temperature, ranging from −24° to −2°C. In general, the simulated ice crystal mass growth compares well with the laboratory measurements at growth times of less than 15 min. The simulated ice crystal masses are within a 20% relative error of the laboratory measurements. The simulation using the spherical ice growth habit at 25 min is also plotted in Fig. 5 (dashed line). From the figure, ice masses from both laboratory measurements and shape-dependent ice growth model simulations at temperature ranges between −6° and −15°C are significantly larger than those from the spherical ice growth calculations at a given growth time, which indicates the advantages of using shape-dependent ice growth rates.

Fig. 5.
Fig. 5.

The ice crystal mass after 3, 5, 10, 15, 20, and 25 min of growth at temperatures ranging from −2° to −30°C. The initial radius of the ice crystal is 4 μm. The wind tunnel measurements from Takahashi et al. (1991) are plotted with different signs for comparison. The spherical ice growth at 25 min is also shown (dashed line).

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

The ice mass growth in Fig. 5 is similar to what is shown in CL94 and Sulia and Harrington (2011), except at temperatures below −22°C. The differences are caused by the different IGR values used at temperatures below −22°C. As expected, the maximum primary and secondary ice crystal mass growths occur near −15° and −6°C, respectively, which correspond to the asymmetrical ice growth temperature zones. The minima of ice crystal mass growth occur at approximately −4°, −8°, and −20°C, with values that are almost the same as those from the spherical growth simulations (dashed line), indicating an isometric ice growth habit at these temperatures (CL94).

Similarly, Fig. 6 shows the simulated a-axis and c-axis growths with time at each temperature compared with laboratory measurements under different temperatures (Takahashi et al. 1991). As expected, the peak growth of the a axis at a temperature of about −15°C corresponds to a fast dendrite growth habit and the peak growth of the c axis at a temperature of about −6°C is caused by a column-like growth habit. Below −20°C, the ice particle growth is less sensitive to temperature. The simulations capture the temperature dependence of the a-axis and c-axis growth habits and are fairly consistent with laboratory measurements, except at temperatures warmer than about −6°C, which might be caused by the ice crystal density parameterizations used in the 1D ice growth model (Harrington et al. 2013b). In summary, the reliability of the ice growth parameterization is supported by consistency between the simulated ice crystal mass and size growth and laboratory measurements.

Fig. 6.
Fig. 6.

As in Fig. 5, but for the (a) a-axis and (b) c-axis length growth with time.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

In addition to comparing the simulations with laboratory measurements, the long-term ground-based measurements of the radar Doppler velocity and Ze also provide reliable evaluations of the Vt parameterizations and simulated Ze profiles under different CTTs and LWPs. Figure 7 shows the temperature dependence of the mean LWPs of SMCs from 4 years of ground-based microwave radiometer measurements at the ACRF NSA Barrow site. Generally, the LWP values decrease gradually with CTT from approximately 114 g m−2 at 0°C down to less than 10 g m−2 at −40°C. In this study, except for the LWP sensitivity simulation in section 3d, the CTT-dependent mean LWP values of SMCs are used to set the LWP in the 1D ice growth model at a specific CTT. The pressure–temperature relations from U.S. Standard Atmosphere, 1976 models (COESA 1976) are used to provide pressure information in the 1D ice growth model.

Fig. 7.
Fig. 7.

Mean LWP values (asterisks) of SMCs as a function of CTT based on 4-yr ground-based microwave radiometer measurements at the ACRF NSA Barrow site. The boxes represent 25%, 50%, and 75% of the LWP at each CTT.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

Based on the mass distribution hypothesis, Sulia and Harrington (2011) reveal that the ice growth habit is dependent on the initial ice crystal size r. From Eq. (2),
e6
Note that , which indicates that smaller initial crystals have larger relative changes in aspect ratio and produce more irregular habits that ultimately lead to greater mass growth. This indicates a dependence of Ze on the initial size of an ice crystal. Considering that ice crystals are initiated through the freezing of different-sized droplets, we assume that the initial ice crystal size distribution follows a modified gamma distribution (Mace et al. 1998):
e7
where DX is the mode length, α0 is the order and is equal to 2 in this study, and NX is the number of particles per unit of volume per unit of length at the functional maximum. Because we calculate the Ze profiles at a unit Ni (1 L−1), the value of NX does not impact the 1D model simulation. As a result of the dependence of the ice aspect ratio on the initial ice crystal size [Eq. (6)], the ice particle size distribution evolves as the particles grow and fall. The evolution of ice particle size distributions using different initial size distributions are shown in Fig. 6 of Sulia and Harrington (2011). The droplet effective radius ranges from approximately 7 to 12 μm in single-layer nonprecipitating stratiform clouds, which is based on previous studies using in situ measurement and Moderate Resolution Imaging Spectroradiometer (MODIS) remote sensing data (Fleishauer et al. 2002; Noh et al. 2013). In this study, using a mode effective radius (half of DX) of 10 μm produces the best comparison of ice growth rate and Vt profiles between the 1D model simulations and ACRF MMCR observations.

In the 1D ice growth model, we must assume a vertical air velocity w for calculations. In general, w in midlevel SMCs is relatively small. In polar region low-level SMCs, w is strongly scale dependent. For SMCs, w can be represented by a Gaussian normal distribution with a mean vertical air velocity of approximately 0 m s−1 and standard deviations σ that depend on the temporal or horizontal scale of the measurements. For example, Fan et al. (2011) showed that Arctic SMCs have ≈ 0 m s−1 and σ that ranges from 0.03 to 0.3 m s−1 as the horizontal spacing decreases from 1000 to 100 m. For the comparison with the ACRF MMCR measurements, a σ value of 0.3 m s−1 is used in the 1D model.

Figure 8 shows the comparisons of ice crystal terminal velocities from 1D ice growth model simulations with 4 years of MMCR Doppler velocity measurements of SMCs over the NSA Barrow site at different CTTs. The MMCR Doppler velocity is composed of reflectivity-weighted particle Vt and w. By averaging the 4 years of data, the mean w is close to zero; therefore, the mean MMCR Doppler velocity represents the reflectivity-weighted ice particle terminal velocity. To compare with MMCR Doppler velocity measurements, reflectivity-weighted ice particle terminal velocity is calculated in the 1D ice growth model. Because Vt is primarily dependent on the particle mass, dimension, and shape, an increase of Vt along the fall distance confirms the ice particle growth and fall pattern in the SMCs as described in section 3a. Comparing the values at different CTTs, Vt has slightly higher values around −10° and −20°C and a slightly lower value around −16°C, which are consistent with the spherical and plate growth habit at those temperature ranges, respectively. In general, the simulated Vt profiles are fairly consistent with the mean MMCR observations at each CTT. In addition, there are significant variations (approximately 25% of the mean) between the Vt and MMCR measurements at each CTT, which might be related to uncertainties in the shape and Vt parameterizations.

Fig. 8.
Fig. 8.

Comparisons of reflectivity-weighted Vt of ice crystals from 1D ice growth model simulations (black solid lines) with 4-yr statistics of MMCR-measured Doppler velocities at each CTT. The red dashed lines indicate mean values. The red boxes represent 25%, 50%, and 75% of the MMCR Doppler velocities at each distance below the cloud top.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

Within the top 500 m of the SMCs, the particle size is generally smaller than the MMCR wavelength, and scattering is still within the Rayleigh scattering regime. According to the definition of the radar reflectivity factor (mm6 m−3),
e8
where Dm is the melted ice particle diameter and f(Dm) is the normalized size distribution in terms of the melted ice particle diameters. The value of Ze is determined by the Ni and ice particle size distributions, and it is more sensitive to ice particle size distributions because of the sixth-power relationship. Based on the ice growth and fallout trajectory in the SMCs described in section 3a, ice crystals are primarily formed at the top layer of SMCs; below that layer, Ni is approximately constant. In situ measurements from the Cloud Layer Experiment (CLEX; Fleishauer et al. 2002), Mixed-Phase Arctic Cloud Experiment (M-PACE; Verlinde et al. 2007), and Indirect and Semi-Direct Aerosol Campaign (ISDAC; McFarquhar et al. 2011) field campaigns confirmed the Ni vertical distribution pattern in SMCs. Therefore, the vertical profile of Ze reflects the gradient of the sixth-power relationship of ice crystal growth. From Eq. (8), for a constant Ni in a vertical column of SMCs, we can define the normalized reflectivity as
e9
where Ze_h (mm6 m−3) is the radar reflectivity factor at a given distance below the cloud top and Ze_r is the reference radar reflectivity factor, which is set to the Ze at 200 m below the top in this study. From Eqs. (8) and (9), Ze_n eliminates the dependence on Ni. Therefore, we can use the observed Ze_n profiles from the radar measurements to evaluate the ice crystal growth along its fallout trajectory that is simulated by the 1D ice growth model.

Figure 9 shows the comparison of Ze_n profiles from 1D ice growth model simulations with 4 years of MMCR measurements over the ACRF NSA Barrow site at different CTTs. In general, the Ze_n increases with the distance below the clouds because of the growth of ice crystals, which is consistent with the ice generation and growth along the fallout trajectory in the SMC conceptual model described in section 3a. Both the MMCR observations and 1D ice growth model simulations using temperature-dependent IGR show fast growth at a CTT of −14° and −16°C, which is primarily caused by the dendrite ice growth habit at these temperatures. The 1D ice growth model simulations that assume a spherical growth underestimate the growth rate statistically at these CTTs. The MMCR observations have approximately ±50% variations at 500 m below the top, which might be caused by MMCR measurements covering a range of LWPs and w at each CTT. In general, the modeled Ze_n profiles are fairly consistent with the MMCR observations at each CTT and provide strong support for the effectiveness of assumptions and parameterization selections in the 1D ice growth model.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the Ze_n profile comparisons. The 1D ice growth model simulations assuming spherical particles (black dashed lines) are also plotted.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

d. Simulated Ze_layer and sensitivity tests

Figure 10 shows the Ze_layer at each CTT calculated from the 1D ice growth model, assuming Ni = 1 L−1 and no vertical air motion and using the mean temperature-dependent LWPs, which are shown in Fig. 7. The value of Ze_layer increases as the CTT decreases to about −6°C and then starts to decrease; it then rapidly increases approximately from −19 to −4 dBZ as the CTT decreases from −10° to −15°C, but it then starts to decrease again steadily. The Ze_layer maxima at approximately −6° and −15°C are caused by column–needle and dendrite growth habits, respectively, whereas the minima around −10° and −20°C are caused by the isometric growth habit. The mean Ze_layer value at each CTT from 4 years of MMCR measurements over the ACRF NSA Barrow site is also plotted. The general pattern of Ze_layer from the 1D ice growth model is similar to the observed mean Ze_layer pattern from the MMCR measurements at warm temperatures (>−20°C), indicating that the 1D ice growth model accurately estimated the temperature-dependent Ze_layer trend. At colder temperatures between −20° and −40°C, the values of the Ze_layer from the 1D ice growth model decrease gradually as the temperature decreases, whereas the values of the Ze_layer from MMCR measurements only decrease slightly. These different Ze_layer patterns indicate a steady increase of Ni as the temperature decreases in the atmosphere. The Ze_layer simulated with the 1D ice growth model without riming growth is also plotted for comparison. At relatively warmer CTTs (>−20°C), the riming growth causes an approximately-2-dBZ-larger Ze_layer. The riming growth is greater around −10°C as a result of the faster falling velocity, which is consistent with laboratory measurements (Fukuta and Takahashi 1999).

Fig. 10.
Fig. 10.

The Ze_layer values calculated from the 1D ice growth model assuming Ni of 1 L−1 (black line) and from 4-yr MMCR measurements over the ACRF NSA Barrow site (red line) at each CTT. The 1D ice growth model simulations without riming growth (blue line) are also plotted to show the relative contributions of diffusional growth and riming growth.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

In the 1D ice growth model, several variables, such as ice crystal density, Vt, and initial size, are parameterized based on previous studies or by using the mean values of variables such as the temperature-dependent IGR and LWP from laboratory and long-term remote sensing measurements, which leads to uncertainties in the simulated Ze_layer. Therefore, we have designed sensitivity tests of the simulated Ze_layer to these parameterizations in the 1D ice growth model. For the baseline simulation, the ice crystal density, Vt, IGR values, initial ice crystal size distribution, and temperature-dependent LWPs are the same as described in section 3c. For the sensitivity tests, one parameterization is changed at a time to quantify the impacts of these parameterizations on the simulated Ze_layer.

Because of the limited laboratory measurements of the IGR, ice crystal densities, and initial ice crystal sizes (especially), it is difficult to discern their uncertainties. However, comparisons of the simulated ice growth rate with laboratory measurements, which are presented above and in Westbrook and Heymsfield (2011), show that the simulated ice growth rates are generally within a 20% error. Figure 11a shows the sensitivity of the simulated Ze_layer to the ±20% uncertainties in ice crystal growth rates. The ±20% uncertainties of the ice crystal growth rates cause less than a 2-dBZ variation in the simulated Ze_layer at each CTT on average. Larger uncertainties of the simulated Ze_layer occur in fast ice growth rate temperature regions (e.g., −15°C).

Fig. 11.
Fig. 11.

Sensitivities of the simulated Ze_layer to parameterizations in the 1D ice growth model: (a) sensitivity to ±20% uncertainties in ice growth rate, (b) sensitivity to ±25% uncertainties in Vt parameterization, (c) sensitivity to and σ, and (d) sensitivity to ±60% LWP variations.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

HW10 showed that the mean errors in their Vt parameterization are less than 25%, relative to the laboratory tank measurements, which are consistent with the observations from the MMCR Doppler velocity measurements as shown in Fig. 7. Figure 11b shows the sensitivity of the simulated Ze_layer to the ±25% uncertainties in the parameterized Vt. From the figure, the ±25% uncertainties of the ice crystal Vt cause an approximately ±3-dBZ variation in the simulated Ze_layer at each CTT. Fukuta and Takahashi (1999) showed that the ice crystal mass is proportional to a growth time to the power (Vt at approximately the − power). Therefore, within the Rayleigh scattering regime, Ze is proportional to a growth time at a power of 3 (Vt at approximately a power of −3). This suggests that an accurate parameterization of Vt is critical for the reliable simulation of the Ze profile from the 1D ice crystal growth model and retrieval of the Ni from the Ze_layer in SMCs.

Because of the high sensitivity of the Ze_layer to the ice particle terminal velocity, w in SMCs may cause considerable variations in the simulated Ze_layer. For an upward w, the ice crystals required additional time to descend 500 m, which generates larger-sized ice particles and a larger Ze_layer, and vice versa. For the collocated CALIPSO and CloudSat measurements, which have a horizontal resolution of approximately 1100 m, is expected to be smaller than 0.03 m s−1 (Fan et al. 2011). For the high-temporal-resolution ACRF ground-based remote sensing measurements, could be larger. Based on 2 years of Doppler radar spectra measurements with a temporal resolution of approximately 4 s over the ACRF NSA site, Chen et al. (2014) showed that the majority of have values within ±0.15 m s−1. To simulate the impact of w, a random number generator that produces normally distributed pseudorandom w for a specific and σ is employed in the 1D ice growth model. Following Fan et al. (2011), a σ value of 0.3 m s−1 is used. Figure 11c shows the impacts of and σ on the simulated Ze_layer. The σ value has a negligible impact on the simulated Ze_layer. However, with values of ±0.15 m s−1 cause an approximately 3-dBZ variation of the simulated Ze_layer at each CTT. Therefore, when applying the retrieval algorithms to high-temporal-resolution ground-based remote sensing measurements of low-level SMCs, retrieved from Doppler radar or lidar can be used as an input in the 1D ice growth model to improve the Ze_layer calculations. When information is not available, the average of multiple profiles over a spatial scale of about 1 km can be used to effectively reduce the impact of .

At a given CTT, the LWP values in the SMCs vary by up to 60%, as shown in Fig. 7. Figure 11d shows the variations of the simulated Ze_layer with ±60% variations (related to the mean) of LWPs at each CTT. In general, the simulated Ze_layer increases with LWP, which is consistent with observations (Zhang 2012). The results show that ±60% variations of the LWPs lead to approximately 3-dBZ variations of the simulated Ze_layer at CTTs warmer than −15°C and less than 2-dBZ variations at colder CTTs. This result indicates a considerable impact of LWP on the simulated Ze_layer. When applying the algorithm to the ACRF ground-based remote sensing measurements, the microwave radiometer (MWR)-measured LWPs are employed as an input in the 1D model. For the CloudSat measurements, the LWPs retrieved from collocated MODIS measurements and/or CALIPSO lidar are used as an input.

4. Ni retrieval and validations

Because of the almost identical normalized ice crystal size distributions in similar SMCs in terms of CTT and LWP (Zhang et al. 2012), it is fair to assume that Ni is the main cause for the Ze differences. From Eq. (8), the Ze (mm6 m−3) for similar SMCs can be expressed as
e10
where is the radar reflectivity factor (mm6 m−3) for normalized ice crystal size distributions at specific CTT and LWP and Ni = 1 L−1. Based on Eq. (10), the Ze differences can be used directly to evaluate the relative magnitude difference of Ni among similar SMCs. Zhang et al. (2012) used this method to quantify the dust impacts on Ni in SMCs over “dust belt” regions and showed that dusty SMCs have 2–6-times higher Ni than the background aerosol condition SMCs at the same CTT and LWP. In this study, by combining the observed Ze_layer values from radar measurements with the values calculated from the 1D ice growth model, which assumes a unit Ni (e.g., 1 L−1), Ni is retrieved from the radar measurements at the specified environmental conditions in terms of CTT and LWP as follows:
e11

Integrated airborne remote sensing observations (such as lidar and radar) and in situ measurements offer an effective method of evaluating remote sensing algorithms (Wang et al. 2012). In this study, integrated airborne remote sensing observations and in situ measurements from several field campaigns, including ISDAC, ICE-L, and the Colorado Airborne Multi-Phase Cloud Study (CAMPS; Chirokova et al. 2011), are used to evaluate the retrieved Ni. The key aircraft for the in situ instruments and measurements used in this study are listed in Table 2. During these field campaigns, the in situ 2D-C probe provides the ice crystal size distributions and their total number concentrations, whereas the integrated airborne radar provides the simultaneous Ze_layer measurements. For the 2D-C probes, shattering effects have the potential to increase the concentrations as an artifact. During the ISDAC campaign, modified tips were used for the 2D-C probe, and artifact removal algorithms were applied to remove the shattering effects (Jackson and McFarquhar 2014). For the two cases that occurred during the ICE-L and CAMPS campaigns, the aircraft flew at approximately 400 m below the cloud top and the maximum particle sizes were generally smaller than 500 μm (Zhang et al. 2012). Furthermore, interarrival times are used to partially remove the shattering effects. Therefore, the shattering effects for these two cases have been significantly reduced (Korolev et al. 2011).

Table 2.

Aircraft in situ instruments and measurements used in this study.

Table 2.

Figure 12 shows an SMC case observed during the CAMPS field campaign on 17 February 2011 over the Rocky Mountains area in Wyoming. The aircraft flew below the liquid-dominated layer. The liquid-dominated layer at the top of the SMC system is observed from the Wyoming Cloud Lidar (WCL) signals (Fig. 12b), and ice crystals below are detected by Wyoming Cloud Radar (WCR) measurements (Fig. 12a). The SMC system has a mean top of approximately 0.5 km above flight level and a mean CTT of approximately −23°C. The correlation between the 2D-C measured Ni and WCR Ze_layer and comparison of the Ni retrieved from the Ze_layer with the 2D-C measurements are shown in Fig. 12c. It is clear that the estimated Ni values are consistent with the 2D-C measurements.

Fig. 12.
Fig. 12.

An SMC system detected during the CAMPS field campaign on 17 Feb 2011. (a) The Ze structure from WCR. The white line near 0.7 km above flight level is the radar-detected cloud top. (b) WCL backscattering power. (c) The 2D-C-measured Ni at flight level (black), WCR-measured Ze_layer (blue), and retrieved Ni (red).

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

Three SMC systems with different CTTs from the ICE-L, ISDAC, and CAMPS field campaigns that are similar to the case presented in Fig. 12 have been selected to validate the retrieved Ni. Approximately 3 h of integrated airborne remote sensing and in situ 2D-C measurements are analyzed. For the ISDAC SMC case on 8 April, the riming growth does not appear to occur, based on a visual judgment of the images of the ice particles, which is likely the result of a low LWC and small droplets (Fan et al. 2011). Thus, riming growth is excluded in the 1D model for the ISDAC case. Figure 13 shows a comparison of the retrieved Ni from the 2D-C measurements for the three SMC systems. The legend on the top-left corner indicates the field campaign name, date, mean CTT (above flight level), respectively. The dotted lines in Fig. 13 denote a factor of 2 lines. The retrieved Ni is consistent with the 2D-C measurements at different Ni and CTTs. Approximately 70% of the retrieved Ni are within a factor of 2, and the mean relative difference between the retrieved and 2D-C measured Ni is about 34%.

Fig. 13.
Fig. 13.

Comparisons of the retrieved Ni with 2D-C measurements for the three SMC systems during ICE-L (black), ISDAC (red), and CAMPS (green) field campaigns. The legend on the top left indicates the field campaign name, date, and mean CTT. The dashed lines are the factor-of-2 lines.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

For simplicity, processes such as cloud-top mixing, liquid droplet contribution to Ze, horizontal wind shear, and aggregation are not considered in the 1D ice growth model for the SMCs. To quantify the impacts of these factors, we evaluate the retrievals with the three-dimensional CRM simulations of an Arctic SMC observed on 10 October 2004 during the M-PACE field campaign over Barrow (Verlinde et al. 2007; Fan et al. 2009). The simulations employed an explicit bin microphysical scheme that uses an aerosol-dependent, temperature-dependent, and supersaturation-dependent ice nucleation scheme based on the parameterization of Khvorostyanov and Curry (2000). A radar simulator is implemented in the CRM to provide online calculations of Ze based on the simulated hydrometeor size distributions. From the CRM simulations, the Ni and simultaneous Ze_layer are obtained. With the CRM-simulated Ze_layer, Ni can be retrieved and compared with the Ni outputs from the three-dimensional CRM simulations.

Figures 14a and 14b show the one-pixel time series of the Ze and Ni profiles, respectively, from the three-dimensional CRM simulations. The mixed-phase stratiform clouds have top heights of about 1.4 km above the surface and CTTs of about −15°C, which correspond to the temperature for the fast ice growth rate of dendrites. Figure 14c shows the time series of the mean Ni within 500 m below the cloud top simulated by the three-dimensional CRM and retrieved from the CRM-calculated Ze_layer, whereas Fig. 14d shows the scatterplot of Ni from the three-dimensional CRM and retrievals over the entire simulation domain. To retrieve Ni, = 0 m s−1, σ = 0.1 m s−1, and CRM-simulated LWP are used in the 1D ice growth model (Fan et al. 2009). The retrieved Ni is fairly consistent with the value from the CRM simulations over a three-order-of-magnitude range of Ni values and generally within a factor of 2, although the retrievals show a slight overestimation at high Ni values.

Fig. 14.
Fig. 14.

Comparison of the retrieved Ni from the three-dimensional CRM-simulated Ze_layer with Ni in the model based on an M-PACE case. (a) The Ze profiles from the radar simulator in the three-dimensional CRM, (b) Ni profiles from the three-dimensional CRM simulation, (c) time series of the mean Ni within 500 m of the cloud top from the three-dimensional CRM simulations and Ze_layer retrievals, and (d) comparison of Ni between the three-dimensional CRM simulations and Ze_layer retrievals. The black dashed lines are the factor-of-2 lines.

Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0354.1

5. Summary

In this study, an algorithm is developed to retrieve ice number concentrations in SMCs from Ze measurements by taking advantage of the simple ice generation and growth pattern. To remove the impact of the temperature-dependent ice growth rates and habits on Ze, a 1D ice growth model is developed to simulate ice diffusional growth and riming growth along the fallout trajectory under different CTTs and LWPs. The simulated ice mass and dimensions with time are compared with laboratory measurements and found to be consistent. The calculated normalized radar reflectivity and ice particle terminal velocity profiles are evaluated with 4 years of ACRF NSA ground-based radar reflectivity and Doppler velocity measurements. The long-term ground-based radar measurements also have provided assistance in selecting the appropriate parameters (e.g., initial sizes) in the 1D ice growth model. Because of the high SMC occurrence and the available updated high-vertical-resolution radar measurements at the NSA site, only NSA radar measurements are used here to validate the 1D ice growth model simulations. However, there are no differences in ice depositional and accretion growth mechanisms between Arctic and low-latitude SMCs. Therefore, the retrieval algorithm should be applicable to SMCs globally.

Sensitivity tests of the simulated Ze_layer to the ice growth rate and ice particle terminal velocity show uncertainties in these parameters that can cause up to 3-dB variations. Therefore, future improvements in the inherent growth ratio, ice crystal apparent density, and fallout velocity parameterizations might further improve the retrieval. The results also show that vertical air motion and LWP uncertainties affect the simulated Ze_layer. Thus, ancillary measurements of w and LWP values in SMCs can be included as inputs to improve the accuracy of the retrieval algorithm whenever they are available. When ancillary measurements of w are not available, temporal- or spatial-averaged radar measurements over a spatial scale of 1 km should be used to ensure mean w is close to 0. In SMCs, the magnitude of LWP impacts the accretion growth rate and the ice growth time under liquid saturation environments. As long as LWP is included as an input in the 1D ice growth calculations and for the retrieval, the retrieval algorithm should be applicable to SMCs with different LWPs. The ice number concentration in SMCs can be retrieved by combining the Ze_layer values from measurements and the 1D ice growth model simulations. To evaluate the retrieved ice number concentration, we analyze the integrated airborne radar and in situ measurements from several field campaigns and compare the retrieved ice number concentration with in situ 2D-C measurements. The retrieved ice number concentrations in SMCs are also evaluated with three-dimensional cloud-resolving model simulations that include more complex microphysical processes. These comparisons statistically show that the retrieved ice number concentrations are within an uncertainty of a factor of 2. However, additional integrated airborne remote sensing observations and in situ aircraft measurements are required to further validate and improve the retrieval in the future.

The algorithm is only applicable to SMCs. Therefore, SMC identification is the first step when implementing the algorithm. Fortunately, the identification of SMCs is straightforward with ground-based or space-borne lidar and radar measurements (Zhang et al. 2010). The results of the study have wide applications. First, the developed 1D ice growth model could be implemented in cloud-resolving models to improve the simulations of SMCs (Harrington et al. 2013a). Second, the algorithm developed in this study could be applied to a large amount of ground-based and space-borne radar measurements to retrieve ice number concentration in SMCs. The long term and global coverage of the datasets enable us to study ice number concentration characteristics in SMCs globally and better understand their geographical variations and dependency on aerosols. Third, the retrieved ice number concentration could be used to evaluate model simulations of SMCs when radar measurements are available.

Acknowledgments

This research was funded by the DOE Grant DE-SC0006974 as part of the ASR program and by the NASA Grants NNX10AN18G and NNX13AQ41G. J. Fan is supported by the DOE ASR program. The ground-based measurement data were obtained from the DOE ARM data archives. The authors thank Jen-Ping Chen, Kara Sulia, and Tempei Hashino for providing the IGR values used in the ice growth model; Robert Jackson and Greg McFarquhar for providing the 2D-C data from ISDAC; and Jeff French and Alfred Rodi for their discussions on techniques in in situ aircraft measurements. Many thanks are also extended to the three anonymous reviewers for their constructive comments.

APPENDIX A

List of Symbols

A Particle cross-sectional area normal to the airflow

a, c Two primary semiaxes of spheroids

C Electrostatic capacitance

Cd Drag coefficient

CTT Cloud-top temperature

D, L Maximum and characteristic dimension of an ice particle

Dm Melted ice particle diameter

dmD, dmR Diffusional and riming mass growths of an ice particle

Dυ Diffusivity of water vapor in the air

E(D, d) Collection efficiency of a droplet d by an ice particle D

Fd Drag force

fυ Ventilation factor

K Thermal conductivity of air

Ld Latent heat of deposition

LWP Liquid water path

N(D) Ice crystal size distribution function

Ni Ice particle number concentration

r Radius of a spherical particle

Re Reynolds number = VtL/νk

Rυ Specific gas constant of water vapor

Sc Schmidt number = νk/Dυ

Si Saturation with respect to ice

T Environmental temperature

V Volume of a spheroid

VD MMCR Doppler velocity

Vt Ice particle terminal velocity

w, , and w′ Vertical air motion, layer-mean vertical air motion, and turbulence-induced vertical air motion

X Best number = 2mD2air/(2)

Z, Ze Radar reflectivity factor with units of mm6 m−3 and dBZ

Zh Radar reflectivity factor at a given distance below cloud top

Zr Reference radar reflectivity factor at 200 m below cloud top

Ze_n Normalized radar reflectivity relative to Zr

Ze_nor Radar reflectivity factor for normalized ice crystal size distribution

Ze_layer Mean Ze between SMC top and 500 m below

α, β Coefficients in the Vt–L relationship

αc, αa Mass deposition coefficients along c and a axes

ϕ Aspect ratio of an ice particle

ρs, ρdep, and ρair Saturated vapor density at T, ice crystal deposition density, and density of air

σ Standard deviation

θe Equivalent potential temperature

Γ Ice inherent growth ratio

νk, η Kinematic and dynamic viscosity νk = ηair

APPENDIX B

Equations for Ice Crystal Diffusional Growth in 1D Ice Growth Model

As described in section 3, a 1D ice growth model is developed to simulate the ice growth trajectory. The ice crystal mass and size evolutions are calculated at each time step (1 s in this study). For nonspherical shapes, radius r is replaced with capacitance C in Eq. (1). Then the mass growth equation is
eb1
Based on CL94, for oblate spheroids (φ < 1), C is given by
eb2
for prolate spheroids (φ > 1),
eb3
The change of ice crystal apparent volume dV from the mass and volume relationship is
eb4
where ρdep is the mass density at the time of deposition. In CL94, ρdep (g cm−3) is expressed as
eb5
where is the excess vapor density (g m−3). Calculated ρdep at given temperatures and after 60 s of growth were shown in Fig. 7 in Miller and Young (1979).
The volume for a spheroid can be expressed as
eb6
and taking the logarithmic differential of Eq. (B6), we get
eb7
The differential of ϕ (=c/a) could be expressed as
eb8
Combine Eqs. (B7) and (B8), and we get
eb9
Therefore, from Eqs. (B4) and (B9), we can obtain the change of ice crystal aspect ratio
eb10
And finally, we can get the changes of the a axis and c axis of the ice crystal,
eb11

APPENDIX C

Terminal Velocity Calculation in the 1D Ice Growth Model

Based on the balance between the drag force of a falling particle and its weight
ec1
where Fd is the drag force, ρair is the density of the air, Vt is the fall velocity, A is the projected area normal to the fall motion, m is the mass of the particle, g is Earth’s gravitational constant, and Cd is the drag coefficient. For calculating terminal velocities, the Best number was employed (Mitchell 1996). Combined with the equation above,
ec2
where D is the maximum dimension of the particle’s projection normal to the direction of fall. Following Böhm (1992), D = 2a for a plate, for a column, and D = 2r for a sphere. Also, η is the dynamical viscosity.
HW10 proposed a modified relationship between drag coefficient and Best number X* based on laboratory data:
ec3
where Ar is the ratio of the particle’s projected area A to the area of a circumscribing circle, Ar = A/[(π/4)D2]. Following Harrington et al. 2013b, for platelike particles and for column-like particles. The variable ρi is ice crystal density; ρbi is bulk density of solid ice (=0.91 g cm−3). Also k has a values ranging between 0 and 1 and is set to 0.5 in this study, following HW10. Given η, ρair, m, Ar, and D, we can calculate the Best number and also the Reynolds number through the X–Re relationship
ec4
where C0 = 0.35, and δ0 = 8.0 for ice crystals. Finally, the terminal velocity is obtained from the equation
ec5

REFERENCES

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