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Problem 2
Part 1
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The figure on the previous page shows logarithmic plots of number concentration N(D), surface area
a(D), mass m(D), and reflectivity favor z(D) as functions of drop diameter (bin) over time. Number
concentration distribution was measured directly by a 2DVD, while the other quantities are directly
proportional to observed D2, D3, and D6, respectively. As such, larger drops have increasingly dominant
contributions to the latter quantities, as discussed below. (Note that the x-axes represent time, but with
units unspecified because they were not available in the dataset).
In the plot of N(D), it can be seen that drops in the smallest bins (< 2 mm) contribute disproportionately
to total number concentration. This is particularly true around t = 50, during the heaviest precipitation,
where number concentration in the smallest bins outnumber the largest bins by a factor of
approximately 104. Excepting the very smallest bins (< 1 mm), N(D) appears to be a strictly decreasing
function of drop size throughout the timeseries.
Both a(D) and m(D) are relatively similar in D and in time. During the heavy precipitation around t = 50,
the peak density for surface area and mass contribution is in the 1-3 mm range, slightly higher than for
N(D). Contributions above 3 mm drop off with D, but less quickly than for N(D). One notable difference
between a(D) and m(D) is seen in the very smallest bin (0.1 mm); it contributes very little to total mass,
as indicated by the dark blue “strip” at the bottom of the distribution.
The reflectivity factor distribution looks nearly opposite to N(D); that is, it appears as an almost strictlyincreasing function of drop size. Bins under 1 mm are virtually inconsequential, with a thicker dark blue
“strip” at the bottom of the plot than was seen for m(D). Beginning at 1 mm, the reflectivity factor
density increases steadily with D, peaking on the order of 105 mm6 m-3 mm-1 for drops in the 5-7 mm
range during the heaviest precipitation.
Part 2
The following plots show timeseries of total number concentration N, total rainwater W, total
reflectivity factor Z, and total rain rate R, respectively. These plots are more similar to one another than
those in part 1. This indicates that, while their respective density functions over D may differ
substantially, these quantities actually mirrored one another fairly well during the event which produced
these observations.
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Part 3
The following plots show how the parameters involved in each DSD model fit vary over time. The DSD
models and the moments used for the fit include, in order: Marshall-Palmer (M3), Exponential (M2,
M4), and Gamma (M2, M4, M6). They include one, two, and three free parameters, respectively.
It should be noted that a unique difficulty was encountered when calculating the Gamma distribution
for some timesteps. These occurred for two cases:
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When µ < -1, the calculation for N = M0 = N0Λ-(µ+1)Γ(µ +1) becomes physically unreasonable due
to the behavior of the gamma function below Γ(0). For this reason, these cases were removed
from the data set (i.e., the Gamma distribution is not computed when this occurs).
When µ becomes large (µ >> 1), the calculation for N0 ~ Λ(µ+3) becomes difficult for the computer
to perform, especially if Λ is also large. Based on empirical testing of the code, a requirement
that µ < 100 was instated, and cases failing to meet this criterion were removed from the
dataset as above.
Marshall-Palmer Distribution (Λ):
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Exponential Distribution (Λ, N0):
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Gamma Distribution (µ, Λ, N0):
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Part 4
The same plots shown in plot 2 are found below; however, the derived quantities from each of the three
DSD model fits are overlaid atop the raw data. For calculating R from the model fits, an empirical
velocity distribution of v(D) = 3.78D0.67 was assumed.
For W, Z, and R, agreement is qualitatively very good between the models and the raw data. The only
discernibly significant error occurs near t = 50, when the Marshall-Palmer (MP) fit briefly diverges from
observed reflectivity (Z). It is unsurprising that this is the case, since MP has the smallest number of free
parameters (one) and reflectivity is proportional to M6, which is farthest from the M3 data used to fit
MP. (By contrast, W and R utilize the very M3 data fed into the MP fit).
For N, agreement between the models and the raw data is not quite as good. Interestingly, the
Exponential and Gamma fits with their additional free parameters are more consistently close to the raw
data, but are also more volatile and feature occasional timesteps in which they exhibit large errors (e.g.,
at the spike near t = 50). The MP distribution is a consistently poorer model, but also never strays into
physically-unreasonable values for N, handling the spike near t = 50 fairly well. The reader is reminded
that gaps in the Gamma distribution curves correspond to data points removed from the dataset, as
described in part 3.
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