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    Quasi-vertical averaged profiles of ZH, ZDR, ρ, and δ in the ML observed with the BoXPol X-band radar in Bonn at 28° elevation on 20 Jun 2013.

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    Relative heights of the (left) ρ and δ extrema, ZH and (center) δ, as well as (right) ZDR and δ in the ML observed with the X-band radars BoXPol in Bonn and JuXPol in Jülich.

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    Correlations between anomalies in the ML observed on 4 Dec 2011, from the 7°-elevation-angle PPI taken by the BoXPol radar in Bonn.

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    PPIs of ZH, ZDR, and ρ observed with BoXPol at 2141 UTC 4 Dec 2011 at 8.1° elevation, showing a dendritic growth signature.

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    PPIs of ZH, ρ, and δ in the ML observed with the S-band KJAX WSR-88D in Jacksonville at 9.9° elevation on 26 Jun 2012.

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    Quasi-vertical averaged profiles of ΦDP, ZH, and ρhυv in the ML observed with the S-band KJAX WSR-88D in Jacksonville, at 9.9° elevation on 26 Jun 2012.

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    Quasi-vertical averaged profiles of ZH, ZDR, ρ, and ΦDP observed with the Essen DWD radar at (top) 12° and at 0908 UTC and (bottom) 24.9° elevation at 0914 UTC 20 Jun 2013.

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    Quasi-vertical profiles of ΦDP, ZH, and ρ from dual-frequency observations with the KEOX and KEVX WSR-88D S-band radars, and the C-band EEC radar, observed around 2216 UTC 13 Jan 2014.

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    Particle size distributions, velocity distributions, and vertical profiles of radar reflectivity within the (a) upper and (b) lower parts of the ML. The two curves indicate the fall velocities for raindrops (dashed line; Atlas et al. 1973) and for unrimed aggregates (dotted line; Locatelli and Hobbs 1974). [Adapted from Barthazy et al. (1998).]

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    Maximal diameter of raindrop originated from melted snowflake as a function of fall distance from the top of the melting layer for unrimed snow, with a temperature lapse rate of 6.5°C km−1 and relative humidity of 100%.

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    Terminal velocities of raindrops and partially melted snowflakes at the top of the melting layer (bottom curve) and at the levels 200, 400, and 600 m below the top. Unrimed snow aloft, temperature lapse rate of 6.5°C km−1, and relative humidity of 100% are assumed.

  • View in gallery

    Size distributions of raindrops and partially melted snowflakes at the top of the ML (upper curve) and at the levels 200, 400, and 600 m below the top of the ML. The Marshall–Palmer distribution of melted diameters with equivalent rain rate 5 mm h−1 is assumed.

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    Volume fraction of water in a melting snowflake as a function of its equivolume diameter at the height level 400 m below the freezing level. Thin and thick lines are for the absence and presence of accretion, respectively.

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    Conceptual plot illustrating the modification of the size spectrum of partially melted snowflakes via accretion and aggregation. The solid line represents the size spectrum before and the dashed line the size spectrum after the interaction processes.

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    Backscatter differential phase δ as a function of the melting hydrometeor diameter at S, C, and X bands. Dotted lines indicate dry snow at the top of the melting layer, dashed lines indicate partially melted snowflakes in the middle of the melting layer (400 m below the top), and the solid line is pure water spheroids. The aspect ratio of particles is equal to 0.7.

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Investigations of Backscatter Differential Phase in the Melting Layer

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  • 1 Atmospheric Dynamics and Predictability Branch, Hans-Ertel-Centre for Weather Research, University of Bonn, Bonn, Germany
  • 2 Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma
  • 3 Meteorological Institute of the University of Bonn, University of Bonn, Bonn, Germany
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Abstract

Backscatter differential phase δ within the melting layer has been identified as a reliably measurable but still underutilized polarimetric variable. Polarimetric radar observations at X band in Germany and S band in the United States are presented that show maximal observed δ of 8.5° at X band but up to 70° at S band. Dual-frequency observations at X and C band in Germany and dual-frequency observations at C and S band in the United States are compared to explore the regional frequency dependencies of the δ signature. Theoretical simulations based on usual assumptions about the microphysical composition of the melting layer cannot reproduce the observed large values of δ at the lower-frequency bands and also underestimate the enhancements in differential reflectivity ZDR and reductions in the cross-correlation coefficient ρ. Simulations using a two-layer T-matrix code and a simple model for the representation of accretion can, however, explain the pronounced δ signatures at S and C bands in conjunction with small δ at X band. The authors conclude that the δ signature bears information about microphysical accretion and aggregation processes in the melting layer and the degree of riming of the snowflakes aloft.

Corresponding author address: Dr. Silke Trömel, Auf dem Hügel 20, 53121 Bonn, Germany. E-mail: silke.troemel@uni-bonn.de

Abstract

Backscatter differential phase δ within the melting layer has been identified as a reliably measurable but still underutilized polarimetric variable. Polarimetric radar observations at X band in Germany and S band in the United States are presented that show maximal observed δ of 8.5° at X band but up to 70° at S band. Dual-frequency observations at X and C band in Germany and dual-frequency observations at C and S band in the United States are compared to explore the regional frequency dependencies of the δ signature. Theoretical simulations based on usual assumptions about the microphysical composition of the melting layer cannot reproduce the observed large values of δ at the lower-frequency bands and also underestimate the enhancements in differential reflectivity ZDR and reductions in the cross-correlation coefficient ρ. Simulations using a two-layer T-matrix code and a simple model for the representation of accretion can, however, explain the pronounced δ signatures at S and C bands in conjunction with small δ at X band. The authors conclude that the δ signature bears information about microphysical accretion and aggregation processes in the melting layer and the degree of riming of the snowflakes aloft.

Corresponding author address: Dr. Silke Trömel, Auf dem Hügel 20, 53121 Bonn, Germany. E-mail: silke.troemel@uni-bonn.de

1. Introduction

The backscatter differential phase δ (°) is defined as the contribution of backscattering from objects within the radar resolution volume to the difference between the phases of horizontally and vertically polarized components of the wave—the total differential phase ΦDP (°)—observed at the radar. Accurate rainfall measurements based on specific differential phase KDP rely on the propagation component of ΦDP and thus require an effective filtering of the backscattered component (e.g., Matrosov et al. 1999, 2002; Otto and Russchenberg 2011; Schneebeli and Berne 2012). Until recently, this troublesome characteristic of δ has been the primary target of researchers while its potential utilization (e.g., for hydrometeor and process typing) has not been well explored.

For Mie scatterers in the radar volume, δ is a tell-tale sign and bears information about the dominant size of drops in rain and of wet snowflakes in the melting layer (ML). Several studies suggest that there is a relation between δ and ZDR—the differential reflectivity (difference between reflectivities at horizontal and vertical polarizations); for example, Otto and Russchenberg (2011) suggested a best-fit relationship:
e1
with δ and ZDR (dB) based on scattering computations at X band. Schneebeli and Berne (2012) confirmed these general findings but came up with the relation
e2
and attributed the difference between both ZDRδ relations to temperature effects. Trömel et al. (2013) analyzed the temperature impact on the ZDRδ relationship based on simulations with distrometer datasets compiled in Oklahoma and Bonn, Germany. They were able to attribute the overwhelming part of the variability unexplained by ZDR to raindrop temperature variation, while drop size distribution (DSD) variability seemed to be of secondary importance.

In this study, we focus on the potential information content of δ in mixed-phase conditions. Berenguer and Zawadzki (2009) report correlations between brightband intensity and ZDR near the surface, which hints at big melting snowflakes in the ML creating big raindrops below. These findings suggest that δ and ZDR measurements and the analysis of their relationship in the ML may open a new avenue for improved ZR relationships (where Z and R represent the linear radar reflectivity and rain rate, respectively) for utilization near the ground. For example, for a given reflectivity in the rain layer, the presence of rimed snow in the ML is manifested by lower ZDR and δ both in the ML and in the rain below relative to unrimed snow (e.g., Ryzhkov et al. 2008). Since δ increases with the dominant size of raindrops or wet snowflakes and its vertical profile thus hints at microphysical processes leading to their formation, the quantification of δ together with other polarimetric variables in the ML might allow for a better microphysical characterization of the bright band. Such information can then be exploited for both improving microphysical models of the ML and quantitative rainfall estimation. An improved ML characterization including its temporal evolution could also be utilized in object-based approaches to precipitation system analysis, which use the ML evolution for system identification and prediction (e.g., Trömel et al. 2009; Trömel and Simmer 2012).

In section 2, the method recently introduced by Trömel et al. (2013) to reliably measure δ in the ML is summarized. In section 3, we report on recent analyses of δ in the ML, together with reflectivity at horizontal polarization ZH, differential reflectivity ZDR, and cross-correlation coefficient ρ (where h and υ represent horizontal and vertical, respectively) observed at X band in Germany and at S band in the United States. Dual-frequency observations at X and C bands in Germany and dual-frequency observations at C and S bands in the United States are also compared in order to explore the regional frequency dependencies of the δ signatures. Section 4 presents simulations of the backscatter differential phase δ in the ML and elucidates the potential microphysical processes, which might be responsible for generating the observed values of δ. A summary and outlook are given in section 5.

2. Method

The measured total differential phase ΦDP routinely exhibits characteristic “bumps” within the ML, which may be associated with either δ or nonuniform beam filling (NBF) (Ryzhkov 2007). Within the ML, ρ can vary from 0.8 to 0.97 and variations of ΦDP (caused by reduced ρ) may become so overwhelming that δ cannot be reliably estimated solely from individual radials. Trömel et al. (2013) suggest a reliable method for estimating δ in the ML; they use azimuthal averaging of ΦDP measured at high antenna elevations (>7°) to suppress statistical fluctuations of ΦDP that are attributed to low ρ. Their method allows for better separation of the effects of δ and KDP and for minimization of the impact of NBF. In cases of uniform stratiform precipitation, averaging may extend over all azimuths prior to δ detection along the range. Values of ΦDP just above and below the ML are then connected with a straight line, and the difference between the actual average profile of ΦDP along the range and the straight line is used to derive the maximal azimuthally averaged δ. For more heterogeneous precipitation fields only an azimuthal sector containing subjectively identified uniform brightband characteristics is averaged [see also Trömel et al. (2013) for the required extent of the averaging].

The total differential phase is a sum of δ and the propagation term proportional to the integral , which decreases at higher elevations due to shortening of the length of the ray path within the melting layer proportionally to 1/sinβ, where β is the antenna elevation. The backscatter differential phase δ is not an integral but is a local parameter that does not depend on the length of the ray path within the melting layer. Consequently, with increasing elevations the forward propagation contribution to ΦDP is successively reduced, leading to increasingly “clean” δ estimates. Additionally, NBF effects are smaller at higher elevations and become negligible relative to the magnitude of δ. This has been verified for the cases presented in the following sections, where we estimate the NBF-induced bias of ΦDP from the product of the vertical gradients of ZH and ΦDP according to Ryzhkov (2007) using the formula
e3
In Eq. (3), Ω denotes the radar beamwidth in degrees, which is 1° for the radars used in this study. Our analysis shows that the NBF-related perturbations of the vertical profile of ΦDP can be neglected for antenna elevations higher than 6°–8° even in the cases with the strongest vertical gradients of Z and ΦDP.

3. Observations of δ in the melting layer

a. δ measurements at X band in Germany

A total of 480 snapshots from 13 precipitation events in Germany observed with the polarimetric X-band radars in Bonn (BoXPol) and Jülich (JuXPol) between July 2010 and June 2013 have been analyzed. Every 5 min both radars take 10 plan position indicator (PPI) scans at different elevation angles, including one scan at 37° for the JuXPol radar. Table 1 provides a list of all events considered together with their respective elevation angles and radial resolutions. The azimuthally averaged profiles of ΦDP, ZDR, ρ, and ZH measured by BoxPol at 2140 UTC 20 June 2013 (Fig. 1) indicate that the ML is at around 3.2-km height by increased ZH and ZDR and a decreased ρ. The increase of ΦDP across the ML is obviously almost exclusively attributed to δ. According to the method described in section 2, the estimated δ is about 3.6°, while the maximum δ observed during all 13 events is about 8.5°.

Table 1.

Radar wavelength, name, date, radial resolution, and elevation angle of the PPIs analyzed in Germany and the United States (single-frequency observations only). Information about the dual-frequency observations is provided in Table 5, below.

Table 1.
Fig. 1.
Fig. 1.

Quasi-vertical averaged profiles of ZH, ZDR, ρ, and δ in the ML observed with the BoXPol X-band radar in Bonn at 28° elevation on 20 Jun 2013.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

For all events the relative heights and magnitudes of the polarimetric moments in the ML have been investigated. The polarimetric theory of the melting layer as described by Giangrande (2007), Ryzhkov et al. (2008), and Trömel et al. (2013) predicts the δ maximum above the ZDR maximum (which is confirmed by our observations; see Fig. 2). The theory is, however, not clear regarding the relative heights of the δ maximum and the ρ minimum. Our X-band observations position the δ maximum above the ρ minimum. The observations show also that the height levels of maximum ZH and δ approximately coincide, which is unexpected. The S-band observations (see section 3b), however, are in agreement with theoretical simulations concerning the relative heights of maximum ZH and δ. These discrepancies have to be further explored.

Fig. 2.
Fig. 2.

Relative heights of the (left) ρ and δ extrema, ZH and (center) δ, as well as (right) ZDR and δ in the ML observed with the X-band radars BoXPol in Bonn and JuXPol in Jülich.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

To investigate the relationship between different polarimetric variables in the ML, the anomalies ΔZH, Δρ, ΔZDR, and δ for all events are determined and correlated with each other. The method introduced by Trömel et al. (2013) is applied to generate azimuthally averaged quasi-vertical profiles based on single PPI scans measured at high antenna elevations prior to this analysis. The anomalies are determined as the difference between the maximal values of the respective polarimetric variable in the ML and the one in rain immediately below the bottom of the ML. The bottom of the ML is defined by the intersection of the radial profile with its straight-line fit. The scatterplots in Fig. 3 summarize the magnitudes of the anomalies observed on 4 December 2011. Note that the maximal Δρ is around −0.2 and maximal ΔZDR is 1.5dB, which is close to the maximal anomalies Δρ = −0.23 and ΔZDR = 1.7dB observed during all 13 of the precipitation events investigated. Both absolute values are higher than expected from simulations, which ignore aggregation and accretion processes (Trömel et al. 2013; Ryzhkov et al. 2008). Table 2 compares Pearson correlation coefficients between the different anomalies in the ML based on all events with the correlations based on the snapshots observed on 4 December 2011, only. The magnitudes of δ and ΔZH in the ML are, as expected, not significantly correlated both on average (left side of Table 2) and for individual cases (e.g., right side of Table 2 and Fig. 3), because δ does not—contrary to ZH—depend on particle concentration. A stronger—negative—correlation exists between δ and Δρ, which is however dominated by only a few events like the 4 December 2011 case (see also Trömel et al. 2013). Thus, the generally weak correlation of δ with other polarimetric variables suggests independent information about the ML carried by δ, which may hint at specific microphysical processes.

Fig. 3.
Fig. 3.

Correlations between anomalies in the ML observed on 4 Dec 2011, from the 7°-elevation-angle PPI taken by the BoXPol radar in Bonn.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Table 2.

Pearson’s correlation coefficients r between the anomalies of observed polarimetric moments in the ML based on all events observed in Germany at X band and based on observation on 4 Dec 2011, only.

Table 2.

Larger aggregates above the ML are generally expected to produce larger δ. Hence, in the absence of riming, for example, associated with higher cloud tops (i.e., deeper subfreezing parts of the storm), intense dendritic growth, indicated by the presence of a fake bright band aloft, makes subsequent aggregation more likely. Thus, a link between intense dendritic growth and higher δ values may exist. No clear correlation between δ and the depth of the cloud has, however, been identified so far. Some link may exist, however, between an intense dendritic growth zone aloft and δ within the ML. Dendritic growth is often observed in the region near −15° along with a sharp vertical increase in ZH in this region and local maxima in KDP and ZDR in stratiform clouds (e.g., Kennedy and Rutledge 2011; Bechini et al. 2013). Below the layer of dendritic growth, ZH increases toward the surface while KDP and ZDR decrease due to aggregation. The signatures above the freezing level observed in several of the investigated cases in Bonn can be identified as the layers of dendritic growth. Figure 4 shows the composite PPI of ZH, ZDR, and ρ observed with BoXPol at 2141 UTC 4 December 2011 at 8.1° elevation. The first layer of enhanced ZDR and reduced ρ associated with a high vertical gradient of ZH above the freezing level is likely the area of dendritic growth. Although the connection between the presence of the layer of pronounced dendritic growth aloft and the δ enhancement within the ML seems to be present in some cases, it is not always the case.

Fig. 4.
Fig. 4.

PPIs of ZH, ZDR, and ρ observed with BoXPol at 2141 UTC 4 Dec 2011 at 8.1° elevation, showing a dendritic growth signature.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

b. δ measurements at S band in the United States

The data for 13 precipitation events spanning both warm and cold seasons of the year observed at S band with the Weather Surveillance Radars-1988 Doppler (WSR-88Ds) in the United States were analyzed (Table 1). Figure 5 shows as an example of the PPIs of ZDR, ρ, and δ in the ML observed with the Jacksonville, Florida (KJAX), radar at 9.9° elevation on 26 June 2012. The ML is quite pronounced across the entire azimuthal range, and all variables shown (all 360° have been averaged in order to derive the quasi-vertically averaged profiles shown in Fig. 6) clearly indicate the ML. All rain events exhibit well-pronounced δ in the melting layer ranging from 3° to 42° at 9.9° elevation with an average value of about 25°. The values of δ measured at higher elevation angles can be stunningly high in some cases (up to 70°; see Table 3).

Fig. 5.
Fig. 5.

PPIs of ZH, ρ, and δ in the ML observed with the S-band KJAX WSR-88D in Jacksonville at 9.9° elevation on 26 Jun 2012.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Fig. 6.
Fig. 6.

Quasi-vertical averaged profiles of ΦDP, ZH, and ρhυv in the ML observed with the S-band KJAX WSR-88D in Jacksonville, at 9.9° elevation on 26 Jun 2012.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Table 3.

Maximal reflectivities Zmax and anomalies ΔΦDP in the ML for different elevation angles observed with the KJAX radar in Jacksonville on 26 Jun 2012.

Table 3.

Such high values of δ measured in the melting layer at S band are quite surprising at first glance and considerably exceed the theoretical expectations put forward in Trömel et al. (2013) that were based on the model of ML without accounting for accretion/aggregation (Szyrmer and Zawadzki 1999; Battaglia et al. 2003; Zawadzki et al. 2005;). In all 13 of the events examined at S band, the maximum of δ coincides very well with the minimum of ρ. The corresponding minimum ρ varies between 0.87 and 0.96 and does not show an obvious correlation with the magnitude of the δ maximum. The δ maximum–ρ minimum is always below the maximum of ZH and above the maximum of ZDR in the S-band observations. Again, as in the X-band cases, we did not find any pronounced correlation between the magnitude of the δ maximum and the maximal value of ZH and its enhancement in the ML. In approximately half of the examined cases in the United States, the increase in δ with increasing elevation angle within the interval from 6.0° to 19.5° was documented. Examples of these elevation dependencies are shown in Tables 3 and 4 for the cases in Florida and Oklahoma. Beam smearing is the most likely explanation for this effect. Indeed, for the height of the ML at 4.6 km in the Florida case, the width of the radar beam (or vertical resolution of the radar) is about 0.75 km at 6.0° elevation angle and more than 3 times smaller at 19.5° elevation angle. The melting layer is usually a few hundred meters deep and the layer of enhanced δ inside the ML is shallower by a factor of 3–4. Hence, an inevitable degradation of the δ signature occurs for poorer vertical resolutions at lower antenna elevations. This is consistent with the decrease of the maximal value of ZH measured in the ML at lower elevations (see Tables 3 and 4).

Table 4.

Maximal reflectivities Zmax and anomalies ΔΦDP in the ML for different elevation angles observed with the KTLX radar in Oklahoma City on 30 Jul 2013.

Table 4.

c. Dual-frequency observations

Since all analyzed X-band observations are from Germany and all S-band observations are from the United States, part of the observed differences between the X and S bands may be attributed to climatic differences. Thus, dual-frequency observations of δ in individual storms have been included for both regions in order to corroborate the unexpected high δ observations at longer wavelengths (Table 5).

Table 5.

Radar wavelength, name, date, radial resolution, and elevation angle of the PPIs analyzed in Germany and the United States for the dual-frequency observations.

Table 5.

In Germany, dual-frequency observations are obtained at the X and C bands for single events observed with BoXPol and JuXPol and adjacent C-band radars from the German Weather Service (Deutscher Wetterdienst, or DWD) network. On 22 June 2011, thunderstorms occurred ahead of a cold front in the region observed by the radars. The BoXPol observations in the stratiform region following the front show δ values around 5° while somewhat smaller δ values around 3° and below are observed by the C-band radar in Essen in accordance with the expectations of Trömel et al. (2013). On 1 November 2013, however, a day with warm-air advection and large-scale rainfall, δ observations at X band around 3° at 28° elevation angle approximately collocate with δ of 25° at 25° elevation angle at C band. At 12° elevation angle the Essen radar even showed δ magnitudes of around 35°.

The most impressive German event analyzed, however, occurred during 20 June 2013. A prefrontal convergence line generated high instability and heavy thunderstorms, which caused a host of flooded basements and ended a heat wave in Germany. Intense lightning activity was reported on 20 June 2013; the international LINET lightning detection network (Betz et al. 2009) measured 854 872 lightning strikes across Germany. Observations from the DWD radar in Essen at 12° and 24.9° elevation are compared now with BoXPol and JuXPol observations at 28°. Figure 7 shows the quasi–vertically averaged profiles of ZH, ZDR, ρ, and ΦDP observed with the Essen DWD radar at 12° elevation (top) and 24.9° elevation (bottom) at 0908 and 0914 UTC, respectively. At 12° elevation angle δ reaches 60° and at 24.9° elevation angle is reaches 84°. These large δ values remained stable for at least 35 min. BoXPol and JuXPol, however, show very small δ values between 1° and 4° during the same time period.

Fig. 7.
Fig. 7.

Quasi-vertical averaged profiles of ZH, ZDR, ρ, and ΦDP observed with the Essen DWD radar at (top) 12° and at 0908 UTC and (bottom) 24.9° elevation at 0914 UTC 20 Jun 2013.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Similar dual-frequency observations have been performed with a C-band polarimetric radar and two operational WSR-88D S-band radars in the United States. Figure 8 compares the quasi-vertical profiles of ΦDP, ZH, and ρ from the WSR-88D S-band radar at Fort Rucker, Alabama (KEOX); the WSR-88D S-band radar at Eglin Air Force Base (AFB), Florida (KEVX); and the C-band radar at the Enterprise Electronics Corporation (EEC) in the city of Enterprise, Alabama, observed around 2216 UTC 13 January 2014. The elevation angles range from 14.6° to 15.6°. It is interesting that although the ZH enhancements within the ML are very similar for the S- and C-band radars, the C-band ρ is much lower than ρ at S band, whereas the C-band δ is almost invisible.

Fig. 8.
Fig. 8.

Quasi-vertical profiles of ΦDP, ZH, and ρ from dual-frequency observations with the KEOX and KEVX WSR-88D S-band radars, and the C-band EEC radar, observed around 2216 UTC 13 Jan 2014.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

4. Theoretical simulations of the backscatter differential phase in the melting layer

The model of the ML without accretion/aggregation, which was used in a number of previous studies (Szyrmer and Zawadzki 1999; Battaglia et al. 2003; Zawadzki et al. 2005) and in the recent paper by Trömel et al. (2013), suggests values of δ below 1°, 3°, and 5° at S, C, and X bands, respectively (see Fig. 8 in Trömel et al. 2013). The model also obviously does not reproduce the observed large values of differential reflectivity ZDR enhancement and the cross-correlation coefficient ρ reduction in the ML (Ryzhkov et al. 2008). The model assumes that snowflakes gradually turn into raindrops by melting. While the density of the melting snowflake increases with fall distance, its size decreases. This assumption creates the largest particles at the very top of the melting layer and contradicts the observations by Stewart et al. (1984), Willis and Heymsfield (1989), Barthazy et al. (1998), and Goeke and Waldvogel (1998), which position the maximal particle size within the ML. This is illustrated in Fig. 9, adapted from the study of Barthazy et al. (1998), who measured the maximal size of snowflakes of 7 mm with an optical spectrometer at the top of the ML. In the middle of the ML or slightly below the maximum of ZH, the maximal measured particle size reaches 23 mm and then rapidly decreases toward the bottom of the ML.

Fig. 9.
Fig. 9.

Particle size distributions, velocity distributions, and vertical profiles of radar reflectivity within the (a) upper and (b) lower parts of the ML. The two curves indicate the fall velocities for raindrops (dashed line; Atlas et al. 1973) and for unrimed aggregates (dotted line; Locatelli and Hobbs 1974). [Adapted from Barthazy et al. (1998).]

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

The further increase of particle size, when falling below the melting point can be explained and eventually modeled by the following mechanisms. Large partially melted snowflakes generated at the top of the ML collect smaller liquid drops originating from smaller completely melted snowflakes below via accretion. Aggregation occurs if partially melted snowflakes collect other partially melted snowflakes. Both processes are believed to be dominant in the upper part of the ML (Goeke and Waldvogel 1998). As the size and water content of large wet snowflakes increase, the snowflakes become unstable and prone to spontaneous breakup and breakup caused by collisions. The processes of accretion, aggregation, and breakup can be described by the stochastic collection equation (e.g., Mitchell 1988), which is, however, very difficult to solve within the ML where the microphysical information necessary to parameterize all three processes is very limited. The process of accretion though is relatively easy to model without resorting to solving a full-fledged stochastic collection equation.

We perform theoretical simulations of the process of snow melting using a simple one-dimensional model with spectral (bin) microphysics, as briefly described in the appendix. The model explicitly treats the process of snow melting by taking into account the initial size distribution and density of dry snowflakes and describes the evolution of the mass water fraction, density, terminal velocity, and size distribution of hydrometeors as the snowflakes fall through the melting layer separately for 80 size bins. Vertical profiles of mass water fraction are derived for every size bin using Eq. (A1) (see the appendix). Smaller-size snowflakes are melted completely near the top of the melting layer and converted into liquid drops, which coexist with larger-size partially melted snowflakes at any given level within the ML. The falling distance of a snowflake before it completely melts depends on its original size, density, temperature lapse rate, and humidity. The density of snowflakes at the top of the ML is assumed to depend on the diameter of the snowflake Ds and the degree of riming frim (Brandes et al. 2007; Zawadzki et al. 2005):
e4
In Eq. (4), ρs is expressed in grams per cubic centimeter and Ds is in millimeters. The degree of riming frim changes from 1 in the case of unrimed snow to 5 for heavily rimed snow aloft (Zawadzki et al. 2005).

The dependence of the maximal size of liquid drops on the fall distance from the top of the melting layer is illustrated in Fig. 10 for the case of unrimed snow (frim = 1), temperature lapse rate of 6.5°C km−1, and relative humidity of 100%. The terminal velocities of hydrometeors are very different for rain and snow with raindrops falling much faster. Figure 11 shows that at 400 m below the top of the melting layer, where the maximal size of raindrops is about 2 mm, the difference between terminal velocities of raindrops Ur and wet snowflakes Us can be as large as 5 m s−1, which favors accretion of partially melted snowflakes.

Fig. 10.
Fig. 10.

Maximal diameter of raindrop originated from melted snowflake as a function of fall distance from the top of the melting layer for unrimed snow, with a temperature lapse rate of 6.5°C km−1 and relative humidity of 100%.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Fig. 11.
Fig. 11.

Terminal velocities of raindrops and partially melted snowflakes at the top of the melting layer (bottom curve) and at the levels 200, 400, and 600 m below the top. Unrimed snow aloft, temperature lapse rate of 6.5°C km−1, and relative humidity of 100% are assumed.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

At any given height level, raindrops originating from fully melted snowflakes and partially melted snowflakes have very distinct size distributions, as Fig. 12 demonstrates. Size distributions shown in Fig. 12 are calculated assuming that the size distribution of melted diameters of snowflakes is of Marshall–Palmer type corresponding to a rain rate of 5 mm h−1. It is clear from Fig. 12 that the size distribution of hydrometeors is close to exponential only at the top and bottom of the ML. Inside the ML, it is close to biexponential, which agrees with the observations of Barthazy et al. (1998) and Goeke and Waldvogel (1998).

Fig. 12.
Fig. 12.

Size distributions of raindrops and partially melted snowflakes at the top of the ML (upper curve) and at the levels 200, 400, and 600 m below the top of the ML. The Marshall–Palmer distribution of melted diameters with equivalent rain rate 5 mm h−1 is assumed.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

The number of raindrops that a large snowflake with equivolume diameter Ds collects during the time interval dt is equal to
e5
where E is the collision efficiency (maximal value is 1); Dr is the equivolume diameter of raindrops; Us and Ur are the terminal velocities of wet snowflakes and raindrops, respectively; N(Dr) is the drop size distribution; and Dr(max) is the diameter of the largest liquid drop, which is a function of the height h below the freezing level (see Fig. 10). During the time interval dt, the snowflake falls a distance dh = Us dt and the total volume of accreted water is
e6
The total accreted water ΔVa that the snowflake collects after falling the distance H down from the freezing level is equal to
e7
During its fall through the layer H, the snowflake partially melts and its volume of water is given by
e8
where fm is the volume water fraction in the absence of accretion/aggregation. Therefore, the total volume of water the snowflake holds is the sum of ΔVa and ΔVm so that the volume fraction of water after accretion is given by
e9
According to Eq. (6), the amount of accreted water depends on the size distribution of liquid drops and the difference between the terminal velocities of raindrops and partially melted snowflakes.

The size dependencies of the volume water fraction on the diameter of melting snowflakes at 400 m below the freezing level were computed using the microphysical model and Eqs. (6)(9) (Fig. 13) for the absence (thin line) and presence (thick line) of accretion. Following Goeke and Waldvogel (1998), it was assumed that the collection efficiency E is equal to 1 in the upper part of the melting layer. At 400 m, the maximal size of liquid drops is about 2 mm (Fig. 10). Beyond this size, the volume water fraction decreases rapidly with increasing size of partially melted snowflakes. Accretion redistributes the water fraction across the size spectrum from the lower end toward its higher end. In other words, larger snowflakes get more water at the expense of the “liquid” part of the spectrum. Although accretion makes larger snowflakes wetter, it does not significantly change their size. The increase of the maximal size of the snowflakes occurs due to aggregation when two large partially melted snowflakes collide and stick together. The combined effect of accretion and aggregation reduces the concentration of smaller-size particles and boosts the concentration of larger-size snowflakes, as the conceptual plot in Fig. 14 shows.

Fig. 13.
Fig. 13.

Volume fraction of water in a melting snowflake as a function of its equivolume diameter at the height level 400 m below the freezing level. Thin and thick lines are for the absence and presence of accretion, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

Fig. 14.
Fig. 14.

Conceptual plot illustrating the modification of the size spectrum of partially melted snowflakes via accretion and aggregation. The solid line represents the size spectrum before and the dashed line the size spectrum after the interaction processes.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

The increase of the water content of large melting snowflakes tremendously impacts their radar characteristics, including the backscatter differential phase δ. We simulated the size dependencies of δ for dry snowflakes (at the freezing level where melting starts), partially melted snowflakes, and hypothetical giant water drops at S (λ = 11.0 cm), C (λ = 5.45 cm), and X band (λ = 3.2 cm) using the two-layer T-matrix code described in Ryzhkov et al. (2013). The hydrometeors were modeled as oblate spheroids with an aspect ratio of 0.7. For partially melted snowflakes in the middle of the ML, two-layer spheroids with an inner core consisting of snow and an outer layer of water were utilized. The equivolume diameter of the inner spheroid was obtained as
e10
where the volume water fraction f is determined from Eq. (8) and is displayed as a thick solid line in Fig. 13. Note that for snowflakes with diameters 15–25 mm the thickness of the water film at the surface is only about 0.25 mm. However, this is sufficient to dramatically change the backscatter differential phase δ, as shown in Fig. 15.
Fig. 15.
Fig. 15.

Backscatter differential phase δ as a function of the melting hydrometeor diameter at S, C, and X bands. Dotted lines indicate dry snow at the top of the melting layer, dashed lines indicate partially melted snowflakes in the middle of the melting layer (400 m below the top), and the solid line is pure water spheroids. The aspect ratio of particles is equal to 0.7.

Citation: Journal of Applied Meteorology and Climatology 53, 10; 10.1175/JAMC-D-14-0050.1

It is not surprising that δ of dry snowflakes is very close to zero at all three radar wavelengths (dotted lines in Fig. 15). A thin film of water at the surface of a snowflake makes its δ comparable to that of a giant pure liquid drop with the same shape and orientation. The simulated values of δ for larger diameters are consistent with the results of the polarimetric radar observations. Both positive and negative values of δ are possible although the positive cases are certainly prevalent. The observed δ results from the integration over the full size spectrum. It is quite possible that large negative δ within the size range of 1.5–1.6 cm at X band may cancel out the contribution of hydrometeors with positive δ and this may explain why the observed magnitudes of δ at X band are generally smaller than those at C and S bands.

Very high values of the backscatter differential phase δ are likely attributed to the presence of large partially melted snowflakes, which grow by accretion and aggregation within the ML. If the microphysical model of the ML does not include any interaction between particles (i.e., no accretion/aggregation), then it cannot produce the large observed values of δ. It can be shown that the magnitude of δ increases with the amount of accreted water. Hence, microphysical factors favoring accretion help to boost δ. The degree of riming frim has a strong effect on the volume of accreted water and the magnitude of δ. Indeed, as Eq. (6) indicates, the factor |UsUr|/Us decreases with increasing frim due to the decrease of the numerator and the increase of denominator. Therefore, the magnitude of δ can be used for an indirect estimation of the degree of riming of snow above the ML.

5. Summary and outlook

Backscatter differential phase δ within the ML is a reliably measurable parameter, which exhibits high variability. A method recently introduced for estimating δ in the melting layer has been applied to polarimetric radar observations at X band in Germany and S band in the United States. Model simulations that assume spheroidal shapes for melting snowflakes in the absence of accretion and aggregation within the ML yield much lower values of δ than observed, especially at S band (Trömel et al. 2013). Contrary to our expectations, δ observations at S band showed much higher magnitudes than the δ observations at X band. The maximal observed δs at X band are 8.5° and 42° at elevation 9.9° at S band. Single measurements at higher elevation angles (19.5°) exhibit even higher δ values up to 70°. Dual-frequency observations of δ in the same events have been included to verify the unexpectedly high δ observations at longer wavelengths. Measurements from the Essen C-band radar in Germany again show δ signatures around 30° and even 60° in one intense thunderstorm case, while the two strongly overlapping X-band radars provide δ values only around 5°. Dual-frequency observations with a C-band EEC radar and two WSR-88D S-band radars (KEOX and KEVX) also confirm larger δ at S band compared to C band.

Theoretical simulations that do not account for any interactions between particles in the ML are not able to reproduce the results of the observations. The observed very high values of the backscatter differential phase δ are likely attributed to the presence of large partially melted snowflakes. Theoretical simulations using a two-layer T-matrix code and a simple model for the representation of accretion are able to explain the origin of the observed pronounced signatures at S and C bands. To simulate the observed large δ magnitudes, the presence of very large water-coated snowflakes with diameters exceeding 1 cm has to be assumed. The accretion of small liquid droplets originating from smaller size, quickly melted snowflakes by larger, partially melted snowflakes leads to significant increases of the water content of larger snowflakes. Both positive and negative values of δ are possible; large negative δ values are expected at X band for hydrometeor melting diameters of around 1.5–1.6 cm. Since the measured δ result from integrating over the particle size spectrum, large negative δ may compensate to a large extent the contribution of hydrometeors with positive δ, which may partially explain the observed small δ magnitudes at X band compared to C and S bands.

We believe that the effects of nonuniform beam filling are negligibly small at elevation angles exceeding 6°–8°. However, the impact of beam smearing resulting in the degraded vertical resolution of the radar within the melting layer can be quite significant and result in the decrease of the observed δ at lower antenna elevations, which has to be taken into account in the microphysical interpretation of the backscatter differential phase.

In summary, very high values of the backscatter differential phase δ are likely attributed to the presence of large partially melted snowflakes within the ML. If the microphysical model of the ML does not include any interaction between particles (i.e., no accretion/aggregation), then it cannot produce the large observed values of δ. The backscatter differential phase δ varies in a wide range, particularly at S and C bands, and definitely contains very important microphysical information about accretion and aggregation processes in the ML, as well as the degree of riming of the snowflakes above the melting layer.

In the future, measurements of δ can probably be utilized as an important calibration parameter for improving microphysical models of the ML. Larger δ can be associated with larger size aggregates above the ML. Additionally, unrimed snow seems to produce much larger δ than rimed snow; thus, δ may be used to estimate the degree of riming aloft. Some link may exist between the appearance of the zone of dendritic growth aloft and δ within the ML. Signatures for dendritic growth have already been identified in several of the German events shown and need further investigations.

Acknowledgments

The research of S. Trömel was carried out within the framework of the Hans-Ertel-Centre for Weather Research (http://www.herz-tb1.uni-bonn.de/). This research network of universities, research institutes, and the Deutscher Wetterdienst is funded by the Federal Ministry of Transport, Building and Urban Development (BMVBS). Alexander Ryzhkov and Pengfei Zhang were supported via funding from NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072 under the U.S. Department of Commerce and from the National Science Foundation (Grant AGS-1143948). We gratefully acknowledge the support of the German Weather Service for providing the radar data for Essen, the support of the Terrestrial Environmental Observatories (TERENO) project funded by the Helmholtz-Gemeinschaft in providing the JuXPol observations, and finally the support of the Transregional Collaborative Research Centre 32 (SFB/TR 32) funded by the German Research Foundation (DFG) for travel funds for A. Ryzhkov to Germany and for providing the BoXPol data. The authors are thankful to the Enterprise Electronics Corporation and to Dr. Qing Cao, in particular, for providing C-band polarimetric data in Alabama.

APPENDIX

Melting of Snowflakes

A one-dimensional model with spectral (bin) microphysics is described, which explicitly treats the process of snow melting by taking into account the initial size distribution and density of dry snowflakes and includes the evolution of mass water fraction, density, terminal velocity, and the size distribution of hydrometeors as they fall through the melting layer separately in 80 size bins. The heat balance equation for a melting snowflake can be written as
ea1
where m is the mass of the melting snowflake, f = mw/m is the mass fraction of melted water, C is the capacitance of the melting snowflake, ka is the thermodynamic conductivity of air, Dυ is the diffusion coefficient of water vapor in air, Lf is the latent heat of fusion (melting), Lυ is the latent heat of vaporization (condensation), Fh and Fυ are thermal and vapor ventilation coefficients, respectively, and TT0 and ρρ0 are the temperature and vapor density differences between the ambient air and the surface of the melting particle. The vapor density difference is expressed as
ea2
where e is the water vapor pressure, es is the saturation water vapor pressure, RH is the relative humidity (%), and Rυ is the water vapor gas constant.
After taking into account that
ea3
where Um is the terminal velocity of a melting snowflake, and assuming that Fh = Fυ = F, we can rewrite Eq. (A1) as an equation for the vertical gradient of the mass water fraction:
ea4
The right-hand side of Eq. (A4) varies in the process of melting. The terminal velocity of a melting snowflake Us depends on its size, density, and mass water fraction. Szyrmer and Zawadzki (1999) and Zawadzki et al. (2005) recommend expressing the terminal velocity of melting snowflakes Us via the terminal velocity of the corresponding raindrops (to which snowflakes melt) Ur and the mass melted water fraction f as
ea5
where
ea6
In addition, Dw is the melted diameter of the snowflake and ρs is the density of the dry snowflake (g cm−3) before melting starts. In our analysis, we use the approximation of Brandes et al. (2002) for the terminal velocity of raindrops:
ea7
In Eq. (A7) ρ is the air density at arbitrary height, ρ0 is the air density at 1000 hPa, Dw is expressed in millimeters, and Ur is in meters per second.
The density of a dry snowflake depends on its equivolume diameter Ds and the degree of riming, frim. In our study, we use the following formula for ρs (Brandes et al. 2007):
ea8
where Ds is expressed in millimeters and ρs is in grams per cubic centimeter. For unrimed snow, frim = 1. If snow is rimed, it is assumed that frim varies between 1 and 5 and does not change across the size spectrum. However, the snow density cannot exceed 0.5 g cm−3 (Zawadzki et al. 2005).
The capacitance C and ventilation factor F depend on the aspect ratio of the particle rm, which is a function of the degree of melting, or the mass water fraction. As in the theoretical studies of Ryzhkov et al. (2013), we assume in our model that the aspect ratio linearly depends on the mass water fraction f as
ea9
In Eq. (A9), rs and rw are the aspect ratios of dry snow particles and raindrops with the same mass. In our model, rs = 0.8 and rw is specified according to Brandes et al. (2002):
ea10
The transformation of the size distribution of the snow particles N(Ds) in the process of melting can be described assuming conservation of mass along the height in each bin of the snowflake melted diameter:
ea11
where N(Dw) is the size distribution of melted diameters (or raindrops that originated from melting snowflakes at the bottom of the melting layer). Taking into account Eq. (A5), the conservation condition Eq. (A11) can be written as
ea12

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