# Analysis of Raindrop Shapes and Scattering Calculations: The Outer Rain Bands of Tropical Depression Nate

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## Abstract

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## 1. Introduction

_{eq}) model by Thurai et al. [4] which is based on direct measurements of drop shapes after 80-m fall. The fundamental oscillation modes and shapes have been summarized in [7]. The dominant mode detected in the wind-tunnel data was the axisymmetric mode with smaller amplitude transverse modes also present. The latter mode has shapes which do not possess an axis of rotational symmetry (henceforth termed as asymmetric shapes) as a result of non-axisymmetric drop oscillations.

_{eq}relations enabled a determination of the importance of asymmetric shapes and their frequency of occurrence relative to the axisymmetric shapes, as in one example of an intense line convection [12,13]. However, in less intense convection or in pure warm rain process (coalescence) dominated events, the frequency of occurrence of asymmetric drops shapes and their impact on Z

_{dr}is not known, which is the subject matter addressed herein. It is well known that small drops dominate the size distributions in tropical rain with active warm rain processes relative to ice dominated deeper convection [14]. It has been speculated that in the ice dominated cases, asymmetric shapes due to oscillations can be dampened by residual tiny ice cores in the nearly fully melted drops (originating as graupel or tiny hail aloft) [15,16]. On the other hand, pure warm rain processes have no such damping mechanism and thus might exhibit more frequent occurrence of asymmetric shapes.

_{eq}> 2.5 mm as well as their differential reflectivities. In Section 6, the computed reflectivity and differential reflectivity based on the scattering calculations over a 1-min interval are compared with the C-band radar measurements over the instrument site, for the entire duration of the TD Nate event. The main conclusions are summarized in Section 7.

## 2. TD Nate Description and Observations in Huntsville

_{eq}= 3.9 mm) is shown in panel (b).

## 3. DVD Data and Processing

_{eq}larger than 1.5 or 2 mm.

_{eq}) did not exceed 4 mm. There were 601 drops with D

_{eq}> 2.5 mm out of 1,467,540 drops in total; out of these, only 79 drops exceeded 3 mm and only 12 drops exceeded 3.5 mm. One of the biggest drops recorded (3.9 mm) is shown in panel (b) of Figure 1 at 14:36:46 UTC. The reflectivity from the C-band ARMOR radar was about 40 dBZ at this time. A small degree of shape deformation is visible.

## 4. Scattering Calculations

_{eq}> 2.5 mm. The simulation program used within this study is CST Microwave Studio (MWS) of the CST Studio Suite 2019. The use of this software for the scattering calculation of each individual drop has been automated. Visual Basic for Application language was used for creating structures and controlling procedures in CST MWS. In the following paragraphs the main steps of the scattering calculations are outlined.

## 5. Results for Individual Particles

_{dr}with drop diameter for all the 601 drops is shown in panel (a) of Figure 5 as cyan color points. For comparison, the theoretical curve (using the T-matrix code) assuming the BC model is included (as the blue dotted line). The zoomed in version for D

_{eq}up to 4 mm is shown in panel (b) for more clarity. In both cases, the cyan points encompass the full range of φ from 0 to 360°. Superimposed on the plots are red points which correspond to φ = 50°, which in fact is close to the look angle (i.e., azimuth) from the ARMOR radar site to the 2DVD location.

- (i)
- Compared to the Z
_{dr}from T-matrix based on the BC model, the Z_{dr}values for individual drops can differ by several dB indicating shape deviations from BC (which is an equilibrium shape model). - (ii)
- The majority of the deviations span both positive and negative values, implying the drop shapes can be ‘elongated’ either in the horizontal plane or along the vertical, due to drop oscillations (including mixed-mode).
- (iii)
- The φ = 50° points show less scatter than the φ = 0 to 360° (cyan) points; this is to be expected.
- (iv)
- The resonance region around D
_{eq}= 6 mm will not have any implications for the outer rain bands of TD Nate since all drops recorded were <4 mm.

_{dr}as function of each drop >2.5 mm during the event. Panel (a) shows the time series of drop sizes. Panel (b) shows the corresponding Z

_{dr}of each drop based on the 2DVD-reconstructed shape and the CST integral equation solver (this is referred to as the calculated Z

_{dr}in panel b). The expected value of Z

_{dr}for each drop is based on the oblate shape axis ratio versus D

_{eq}model of [5]. Panel (c) shows the difference between the calculated and expected values, which can be denoted as δ(Z

_{dr}) = Z

_{dr}[from drop-by-drop 3D-reconstructed shape] minus Z

_{dr}[from drop-by-drop mean oblate shape]. Panel (d, e) show, respectively, the scatter plot of δ(Z

_{dr}) versus D

_{eq}and δ(Z

_{dr}) versus the expected value of Z

_{dr}.

- (i)
- The outer rain bands of TD Nate did not produce drops larger than 4 mm over Huntsville (as mentioned earlier);
- (ii)
- The δ(Z
_{dr}) can be significant, but overall, they tend to be distributed evenly around 0 dB. For the larger drops (>3 mm), δ(Z_{dr}) showed a tendency to be slightly more negative, i.e., drops being somewhat closer to spherical in shape; - (iii)
- The skewness towards negative δ(Z
_{dr}) values were more apparent just prior to 15:00 UTC when the wind speeds were seen to increase and the change in wind direction was more rapid. It was around this time that the fall velocities also showed a small but noticeable number of drops having lower than expected velocities.

_{dr}) and slowing down of the fall velocities (relative to terminal speeds) at the surface. This correlation is speculative as there are no studies linking vertical air motion and riming above the bright-band to drop shapes and fall velocities near the surface.

_{dr}) for D

_{eq}> 2.5 mm for the outer rain bands is shown in Figure 8a. The mode of the distribution lies very close to 0 dB. Significantly more negative δ(Z

_{dr}) were observed, compared with positive δ(Z

_{dr}). To quantify this, as an example, 149 drops out of 601 drops had δ(Z

_{dr}) less than −0.5 dB whereas only 72 drops had δ(Z

_{dr}) greater than 0.5 dB. Recall that δ(Z

_{dr}) is the difference between Z

_{dr}from 3D-reconstructed drop shapes and the Z

_{dr}from oblate axis ratio versus D

_{eq}model given in [5]. For example, consider the extreme case where δ(Z

_{dr}) for all drops were negative, i.e., more spherical on average. Further, if the algorithm for retrieval of D

_{m}(mass-weighted mean diameter) from Z

_{dr}were derived based on expected oblate axis ratio model of Thurai et al. alone, then the D

_{m}would be biased too high for the measured Z

_{dr}, and for a given reflectivity, the rain rates would be biased too low [1]. In our case the mode of the distribution is close to 0 dB but visually the distribution is negatively skewed (Figure 8a). By way of comparison, panel (b) shows the δ(Z

_{dr}) histogram derived from the 80 m fall artificial rain experiment ([4,5]), where only 5% of the measured drops were found to have significant asymmetry. In the latter experiment, the axis ratio of each drop was calculated as the ratio of the vertical chord to the horizontal chord (chord is a line segment joining any two points on a curve) The Z

_{dr}for each drop was then estimated using the relation from Jameson [30]:

_{dr}from (1) is defined as the calculated value. Note that the axis ratio distribution reflects drop oscillations in the 80 m fall experiment. The expected value of Z

_{dr}for each drop is (as before) based on the oblate shape axis ratio versus D

_{eq}model of [5]. The motivation to compare the histograms in this manner is that the 80 m fall experiment can be considered as a baseline for axis ratio distributions for drops <9 mm in very light wind conditions. Whereas the outer bands of TD Nate had D

_{eq}< 4 mm under higher wind conditions (speeds ~8 m/s). It is useful to know that the mode of the histograms is close to 0 dB implying that the most probable axis ratios are not very different between the natural rain event and the artificial rain experiment. The width of the histogram in Figure 8b seems larger than that in Figure 8a which might be due to much larger drops in the artificial rain experiment with correspondingly larger oscillation amplitudes ([4,5]). Finally, the slight negative skewness in Figure 8a indicates a higher proportion of asymmetric drops as mentioned above.

## 6. One-Minute Based Z_{h}, Z_{dr} Calculations and Comparisons with ARMOR

_{i}

^{h}, then the overall reflectivity from all drops over the 1-min period is given by:

_{i}is the vertical velocity of the ith drop. For V polarization, similar integration is performed using the corresponding RCS values, z

_{i}

^{v}. Both are converted to the conventional dBZ units and Z

_{dr}for that 1-min period is determined from the difference between the two.

_{dr}were further smoothed over 3 consecutive points. Panel (a) shows the Z comparison, and panel (b) the Z

_{dr}comparison. After 05:00, there is excellent agreement in both cases. Prior to 05:00, radar Z appears to be somewhat lower than the simulations and the Z

_{dr}slightly higher. It was around this time that the retrieved horizontal drop velocities (from 2DVD measurements) showed some discrepancy with the wind sensor data. Hence it is possible that the shape reconstruction is not precise enough to provide sufficiently accurate scattering amplitudes. It is not possible to consider all the factors that might have led to the discrepancy since simulated reflectivity is overestimated and the Z

_{dr}is underestimated relative to radar measurements (i.e., opposite directions). The dependence of reflectivity on drop shapes should be minor so the simulated Z

_{h}could be more a result of 2DVD-based sizing while the underestimation of Z

_{dr}could be more related to 3D-reconstruction errors.

_{dr}versus Z

_{h}variation from the CST simulations for 05:00–20:00 are compared with those from the radar observations in Figure 10. The variations are in good agreement with each other and both show a clear increase of Z

_{dr}with increasing Z

_{h}beyond 22 dBZ. On average the variation of Zdr versus Z is close to that obtained by Wang et al. [31] based on S-band WSR−88D measurements for stratiform rain for Z > 30 dBZ and closer to tropical rain for Z < 25 dBZ (their Figure 3). This is reasonable since the outer rain bands probably lie between these two rain types. Note also the maximum reflectivity values for both cases [31] did not go beyond 40 dBZ and the maximum Z

_{dr}did not go beyond 1.5 dB. All these features provide confirmation and validation for the drop-by-drop based scattering calculations.

## 7. Conclusions

- (i)
- The horizontal velocity of the drops deduced from the 2DVD is close to the horizontal wind velocity, both in terms of magnitude and direction. This implies that deskewing can be done for drops that do not have an axis of symmetry, using horizontal wind data preferably at the 2DVD sensor height above ground.
- (ii)
- The Z
_{dr}calculated for the 3D-reconstructed drops (drop-by-drop calculations) showed substantial scatter, mainly from the azimuthal variation or look angle. The scatter reduced considerably when the look angle corresponding to the 2DVD camera viewing direction was used. On average, the Z_{dr}from the 3D-reconstructed shapes were in good agreement with the Z_{dr}based on the BC model (though drop-by-drop deviations in Z_{dr}do occur). - (iii)
- The histogram of δ(Z
_{dr}) from the rain bands of TD Nate and the 80-m fall artificial rain experiment were compared. Both histograms had a mode at 0 dB implying that the most probable axis ratios are not very different between the natural rain event and the artificial rain experiment. However, the histogram corresponding to the rain bands of TD Nate was negatively skewed and with smaller width compared to the 80-m fall experiment. The slight negative skewness indicates a higher proportion of asymmetric drops as mentioned above while the smaller width is likely a result of the maximum D_{eq}< 4 mm (compared with D_{eq}< 9 mm for the artificial rain experiment). - (iv)
- The Z and Z
_{dr}calculated using the individual C-band scattering amplitudes of each drop over 1-min interval shows, in general, excellent agreement with the C-band radar observations over the disdrometer for the entire 18 h period of the outer rain bands of TD Nate in Huntsville. On average the Z_{dr}values for a given interval of Z are consistent with rain type being stratiform-tropical.

_{eq}relations (either the BC equilibrium model [2], or [5] or empirical fit given in [34]) together with Gaussian canting angle distribution [35] for simulating polarimetric radar observables. Here, we use a higher order simulation based on scattering from 3D-reconstructed shapes and drop-by-drop integration which is possible with recent advances in 2DVD data processing. In the simulations of Z

_{dr}for the rain bands of TD Nate, the drop-by-drop approach on average gave similar values as using the “bulk” assumptions mentioned above. Several other event analyses have also shown this to be the case [12]. For strong convective rain with large drops, which are associated with high wind speeds and/or rapid change in wind direction, the drop-by-drop approach is preferred over the “bulk” approach as the former accounts for the variance in shapes (the orientation variance is built-in), which are important for other dual-polarization variables such as the copolar correlation coefficient and depolarization ratios [12].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Composite next generation radar (NEXRAD) radar image of reflectivity during Tropical Depression Nate over Alabama on 08 October 2017 at 14:45 UTC. The location of the 2D-video disdrometer (2DVD) and other instruments is marked with a red dot; (

**b**) shape of a large drop recorded by the 2DVD at this time (using the 3D reconstruction procedure, see Section 3).

**Figure 2.**(

**a**) Wind speed and, (

**b**) wind direction (as black points) from the anemometer at 10 m height at the 2DVD location, respectively, along with the retrieved drop horizontal velocities from the 2DVD in red for D

_{eq}> 2 mm averaged over 3 min; (

**c**) percentage change in the drop fall velocities from the 2DVD compared with Gunn–Kinzer.

**Figure 3.**(

**a**) Triangle mesh of an individual drop imported into CST Microwave Studio; (

**b**) histogram of the edge length of the triangulation.

**Figure 4.**Radar Cross Section (RCS) of 601 reconstructed raindrops for horizontal and vertical polarization, as a function of their respective equal volume diameter. For comparison, the RCS of a sphere is shown. f = 5.625 GHz and the refractive index m = 8.6137-j1.3020. Panel (

**a**) is for a fixed view angle of φ = 0, and panel (

**b**) is the averaged value of the RCS over φ = 0 to 359°. Panel (

**c**) shows the same RCS versus D

_{eq}as panel (

**b**) for the reconstructed drops for H and V polarizations, compared with those assuming the BC model is included (solid line, respectively, dashed line are for H,V polarizations).

**Figure 5.**(

**a**) Single drop Z

_{dr}versus D

_{eq}for the reconstructed drops for all φ angles in cyan and for φ = 50° in red, together with those for the Beard–Chuang (BC) model (dashed line in blue) using T-matrix; (

**b**) same as panel (

**a**) but zoomed in to cover the brown dotted line box.

**Figure 6.**(

**a**) Diameters of all drops >2.5 mm recorded during the passage of the outer rain bands of tropical depression (TD) Nate over the 2DVD; (

**b**) (top panel) the calculated Z

_{dr}for each 3D-reconstructed drop in red and the expected Z

_{dr}for each drop (in blue), assuming the oblate axis ratio of Thurai et al., and δZ(

_{dr}) for each particle (bottom panel); (

**c**) δ(Z

_{dr}) versus D

_{eq}; (

**d**) δ(Z

_{dr}) versus the expected Z

_{dr}.

**Figure 7.**Time series of vertical profiles of reflectivity from the vertically pointing XPR observations for 14:24 to 15:00 UTC on 08 October 2017.

**Figure 8.**Histograms of δ(Z

_{dr}) for all >2.5 mm drops (

**a**) for the reconstructed drops from the outer rain bands of TD Nate and (

**b**) for drops recorded during the 80 m fall experiment [4] using Equation (1).

**Figure 9.**(

**a**) Reflectivity and (

**b**) Z

_{dr}calculations derived from the individual drop scattering amplitudes using the 2DVD measurements (in black) compared with the C-band dual-polarization radar (ARMOR) radar measurements (magenta) over the 2DVD site. For the former, 1-min time interval is used.

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**MDPI and ACS Style**

Thurai, M.; Steger, S.; Teschl, F.; Schönhuber, M.
Analysis of Raindrop Shapes and Scattering Calculations: The Outer Rain Bands of Tropical Depression Nate. *Atmosphere* **2020**, *11*, 114.
https://doi.org/10.3390/atmos11010114

**AMA Style**

Thurai M, Steger S, Teschl F, Schönhuber M.
Analysis of Raindrop Shapes and Scattering Calculations: The Outer Rain Bands of Tropical Depression Nate. *Atmosphere*. 2020; 11(1):114.
https://doi.org/10.3390/atmos11010114

**Chicago/Turabian Style**

Thurai, Merhala, Sophie Steger, Franz Teschl, and Michael Schönhuber.
2020. "Analysis of Raindrop Shapes and Scattering Calculations: The Outer Rain Bands of Tropical Depression Nate" *Atmosphere* 11, no. 1: 114.
https://doi.org/10.3390/atmos11010114