Introduction
Present methods of passive and active microwave remote sensing
of precipitation have a key problem: the uncertainty of the physical and
associated radiative properties of ice- and mixed-phase snowflakes. In
nature, ice particles manifest themselves in an extraordinarily diverse
variety of sizes, shapes, and habits – ranging from simple crystals such as
needles or plates to complex aggregates and rimed particles.
Microwave radiation is sensitive to the presence of liquid water on ice-phase
precipitation (e.g., ), due, primarily, to the large difference in
the dielectric constants of the two materials at microwave frequencies.
However, the relationship between the early stages of melting and incident
microwave radiation has not been well described in the literature. With this
in mind, the present research seeks to quantify the sensitivity of microwave
scattering and extinction cross sections to the onset of melting (melt
fractions less than 0.15), with an emphasis on ultimately improving the
forward-model simulation of the physical and radiative properties of
realistically shaped mixed-phase precipitation particles. The onset of
melting is generally believed to represent the most rapid changes in the
scattering and extinction properties of hydrometeors. made
direct measurements and simulation of a normalized radar backscattering
cross section for spongy ice spheres with various liquid water contents. Most
notable is that they show a a rapid change in the backscattering
cross section between 0 and about 7 % liquid water volume fraction.
also noted that the onset of melting represents a
critical point where the radar reflectivity increases rapidly.
A comprehensive examination of the microwave sensitivity to the entire range
of melting is left for future research.
The model described for the first time here, named the Single Particle
Melting Model (hereafter SPMM), is a heuristic model designed to provide
a basis for simulating the physical description of melting individual ice
crystals having an arbitrary shape. By using simple rules and
nearest-neighbor interactions, the melting process is simulated with
a reasonable facsimile of reality. In SPMM, there are no explicit
thermodynamic or physical properties, other than the 3-D shape and the
relative positions of liquid and ice constituents. SPMM is an extremely
computationally efficient algorithm for creating a series of melted particles
ranging from unmelted to completely melted, only requiring several minutes on
a normal desktop computer for a single particle. Thermodynamic melting-layer
models can easily be employed to determine bulk meltwater generation
(), and the SPMM-melted particles can be
mapped into an appropriate particle size distribution according to the
layer-averaged melting properties.
While the general physical properties and thermodynamics of melting
snowflakes are fairly well understood, the complex interaction between
realistically shaped melting snowflake aggregates and incident microwave
radiation has been sparsely examined. Recently, radar properties of melting
aggregates at 3.0 and 35.6 GHz have been simulated
by . There are no known simulations of realistically shaped
melting aggregate hydrometeors at frequencies above 35.6 GHz, for
either passive or active microwave remote-sensing applications. For dry
realistically shaped aggregates, compared simulated
reflectivities from aggregates and soft spheres at Ku, Ka, and the passive
microwave response 18.7, 36.5, and 89.0 GHz, and
has examined the Ku-, Ka-, and W-band (14, 35, and 94 GHz) response to
variations in shape and size, but neither paper consider melting.
In this study, the discrete dipole approximation (DDA), using DDSCAT (Discrete Dipole Scattering) version
7.3 , is employed to compute the scattering and extinction
efficiencies of individual particles with mass Ftmelt fractions ranging from
from 0.0 (unmelted) to 0.15 (lightly melted). Due to the significant
computational requirements of DDSCAT when liquid water is present, we have
limited this study to two selected aggregate particles, shown in
Fig. . The chosen microwave frequencies – 13.4, 35.6, 89.0, 94.0,
165.0, and 183.31 GHz – are relevant to current passive and active
microwave sensors, such as CloudSat , the recently
launched Global Precipitation Measurement mission (GPM; ), and
instrumentation employed during aircraft- and ground-based precipitation
validation experiments over the past decade (e.g., MC3E – ;
GCPEx – ; and IPHEx – ) and the upcoming OLYMPEx
field campaign in winter 2015/16 .
The following sections describe the melting simulation methodology, the
single-particle scattering and extinction properties at standard radar and
radiometer center frequencies, and the particle size distribution averaged
properties with implications for remote-sensing applications.
Selected steps in the onset of melting for (a) needle aggregate
(NA) and (b) dendrite aggregate (DA). Blue regions are ice, and red
regions are liquid water; mass fraction of meltwater is indicated at each
step. Particle mass is conserved throughout the melting process.
Physical description of melting
In an actual melting snowflake, the distribution of water on the surface of
a melting ice crystal is governed primarily by the local amount of liquid
water. According to , in the initial stages of melting,
meltwater distributes more or less evenly over the surface of the ice
crystal. At a locally critical liquid water content, surface tension effects
take over and convex water droplets form at branch points and other
energetically favorable regions on the surface of the underlying ice crystal.
As the crystal shape changes due to additional melting, these water droplets
coagulate and tend to collect toward a common center under the influence of
surface tension. Previous studies have examined the effects of modifying the
distribution of meltwater on oblate spheroids , and one
has explicitly examined melting realistically shaped snowflakes, but with ad
hoc placement of water spheres to simulate melting . In
reality, tumbling, breakup, shedding, collision/aggregation,
evaporation/condensation, and other physical interactions often significantly
alter growth and melting processes. Due to the complexities of such
simulations, we do not explicitly consider these effects in the present
physical model – although aggregation is simulated through the creation of
aggregate snowflakes.
Single Particle Melting Model
The Single Particle Melting Model, developed by the primary author and
described for the first time here, performs physical particle simulations on
an integer-indexed three-dimensional Cartesian grid. Each occupied point
represents a small, finite unit of mass of either ice or liquid. This
assemblage of ice and water points constitutes the particle mass and volume;
Fig. illustrates this for the two selected aggregate shapes used
in this study. In this figure, blue regions represent solid ice and red
regions represent meltwater (red is used for visual clarity). The maximum
melt fraction is 0.15 for the present study, in order to assess the microwave
sensitivity to the initial stages of melting. Throughout this melting range,
casual inspection reveals that very little structural changes occur in either
the dendrite aggregate (DA) or needle aggregate (NA).
The basis for melting and meltwater movement in SPMM occurs through
nearest-neighbor interactions. In a 3-D domain, any given point has 26
nearest neighbors (including all diagonals). The interaction distance is
limited to one neighbor, simplifying the computational requirements of the
algorithm. Figure conceptually illustrates the melting process in
two dimensions; the same logic applies to the three-dimensional model. The
melting simulation proceeds iteratively until all ice is melted and a nearly
spherical droplet is formed.
The SPMM proceeds with the following steps:
Populate a 3-D Cartesian grid with “ice” points, the ensemble of which comprise the entire volume of the simulated snowflake or aggregate.
Iterate over all ice points, tabulating all 26 nearest neighbors (6 sides, 23 diagonals, excluding self).
The ice points having the fewest ice neighbors are melted (see Fig. a).
A stochastic control factor is employed to control the rate of melting.
After each melt iteration, a movement check is applied to those liquid points having zero ice neighbors.
Prohibiting the movement of liquid points that do have ice neighbors simulates a “coating”
effect, whereas liquid points with no ice neighbors are able to move (Fig. b and c).
Movement is a weighted random walk, subject to certain constraints. The walk is weighted
toward total particle
center of mass, simulating the coalescence of liquid water. The movement phase iterates until
all moving liquid cannot move to an open space closer to the center of mass than the current position. Return to step 2.
A simplified diagram depicting the steps in the SPMM. At each step, ice
(blue) points are converted to liquid (red) following nearest-neighbor rules
(see text). Numbers depict the number of nearest ice neighbors. Panel
(a) indicates the first iteration of the melting algorithm;
(b) shows the second iteration and the allowable move conditions;
and (c) shows the third iteration, where more ice has melted and
three points are now free to move into the green regions.
In this simple model, ice structure collapse, breakup, or water shedding are
not explicitly simulated – any orphaned droplets created during melting will
naturally migrate towards the total center of mass as cohesive droplets.
Although the present study considers melt fractions up to 0.15, SPMM provides
melting simulations for any arbitrary particle shape until it is completely
melted. In principle, it can be applied to the melting of any material, where
surface tension is a dominant factor in the liquid phase. The melting
increments and distribution of meltwater can be finely tuned to suit the
application. On a modern desktop computer, a particle having 200 000 ice
points (“dipoles” in DDA parlance) can complete the entire melting process
in less than 5 min using a single processor core. The Fortran 90 codes
and MATLAB codes for SPMM are freely and indefinitely available upon
request from the primary author.
Single-particle scattering and extinction properties
In the previous section we described the method for simulating the melting of
realistically shaped precipitation hydrometeors using SPMM. Given these two
example sets of melted ice particles, the single-particle extinction
cross section, backscattering cross section, and asymmetry parameter
properties were computed at 23 melting steps between 0.0 (unmelted) and 0.15
(lightly melted; see Fig. ). To compute the single-particle
radiative properties, the DDSCAT 7.3 model is used .
Numerical scattering calculations
DDSCAT is an implementation of the DDA method
for solving Maxwell's equations for linearly polarized electromagnetic plane
waves incident upon an arbitrarily shaped dielectric material consisting of
up to nine different dielectric materials; here we only use two discrete
constituent materials: ice and liquid water, neglecting the dielectric
constant of air and water vapor. For satisfactory convergence, DDSCAT
requires that |m|kd<0.5, where |m| is the magnitude of the complex index
of refraction of ice or liquid water , k is the
wave number of the incident radiation, and d is the minimum spacing between
adjacent dipoles. In each of the following calculations, specific care
was taken to ensure that this convergence criterion was adequately satisfied.
In practice this was accomplished by having a sufficiently large number of
dipoles representing the shapes, minimizing the dipole spacing d
compensating for the increase in |m| imposed by the increasing amount of
liquid water present in the melting simulation.
DDSCAT requires the following inputs: the polarization and wavelength of
incident radiation, the 3-D Cartesian position and index of refraction for
each dipole point, the effective radius of the entire particle (aeff: the
radius of a sphere having equal mass), and 3-D rotation angles of the target
in the reference frame. Here, the effective radius (aeff) acts as a proxy for
particle mass, independent of the particle shape.
The outputs of interest for this study are extinction cross section,
scattering asymmetry parameter, and radar backscattering cross section at 30
effective radius intervals – ranging in log scale from 50 microns to
2500 microns. For each effective radius value, the input shape (relative dipole
positions) remain the same, but each dipole mass (and consequently effective
volume) is scaled appropriately. We also examine the extinction and
backscattering efficiencies, Qext and Qbck, which are
simply the respective cross sections (e.g., Cext) divided by the
cross-sectional area of the equivalent sphere (defined in DDSCAT by
Q∗=C∗/πaeff2).
It should be noted that scaling particles
in this manner does not create a mass–dimension relationship that is consistent
with observations when these particles are used together in an ensemble.
Each particle and its associated scattering and extinction properties are intended
to be considered as independent particles if they are being used in an actual
retrieval framework, and it is up to the researcher to ensure that the
mass–dimension relationship they prefer to use is followed.
found that the method of producing aggregates can result in significantly different
optical properties, depending on the ultimate mass–dimension relationship.
in the present study, the purpose of choosing the shape-preserving scaling approach was
so that the the melting morphology would be preserved for a given base shape across
all ranges of masses, enabling a consistent comparison independent of shape changes.
To simulate randomly oriented hydrometeors, an average over multiple
orientations of the aggregate relative to a fixed direction of incident
radiation is computed. This provides an orientation-averaged set of
scattering and extinction properties. Although not shown here, our
sensitivity studies suggest that 75 discrete orientations, sampling a full
3-D rotation, are sufficient to provide a reasonably precise
orientationally averaged set of scattering and extinction calculations. This
trade-off keeps the computational requirements tractable: for a single
effective radius, single shape, and single frequency, it requires
24 h to perform one set of calculations running in a parallel
implementation a 24-core Intel(R) Xeon(R) CPU X5670 at 2.93 GHz and
requires up to 32 GB of allocated RAM per DDSCAT process. For 6
frequencies, 2 shapes, and 30 effective radii, it requires computation times on
the order of 200 days of continuous calculations (with 3 processes
running in parallel). In addition the orientation-averaged quantities, the
scattering and extinction properties are also tabulated for each individual
orientation, which has important implications for exploring the polarization
of scattered radiation, but this is left for future research.
In the current study, the single particle scattering and extinction
calculations are divided into two frequency groups:
Radar-specific frequencies, approximately consistent with the Global Precipitation
Measurement mission Dual-Wavelength Precipitation Radar (GPM DPR) at Ku-band and
Ka-band (13.4 and 35.6 GHz), and with the CloudSat W band radar (94.0 GHz).
Several ground-based and aircraft-based instruments also use these channels for precipitation remote-sensing applications.
“High-frequency” passive microwave channels at 89, 165, and
183.31 GHz, consistent with the GPM Microwave Imager (GMI) channels
used for detecting scattering by ice clouds and ice-phase precipitation.
Similar channels are used on other satellites, aircraft, and ground-based
passive microwave instruments (e.g., SSMI/S, TRMM, CoSMIR) for cloud and
precipitation remote sensing.
Overview of 13.4 GHz extinction and backscattering:
dendrite aggregate (row 1) (a) mean extinction cross section vs.
melt fraction and (b) mean backscattering cross section vs. melt
fraction for various values of effective radius in microns (color bar).
Needle aggregate (row 2) (c) mean extinction cross section vs. melt
fraction and (d) mean backscattering cross section.
Backscattering and extinction at radar frequencies
In Fig. , the 13.4 GHz extinction and radar backscattering
cross sections are computed for the DA in panels (a) and
(b) respectively, and for the NA panels (c) and (d).
Colors represent the particle effective radius (values on the color bar are in
microns), which is directly related to the particle mass. Both extinction and
backscattering cross sections are most strongly influenced by changes in
size. At the smaller sizes, however, the onset of melting has a strong
influence on extinction (panels a and c). This indicates that the onset of
melting is characterized by a rapid increase in extinction, while the
backscattering tends to exhibit a more linear increase. This is consistent
with the integrated backscattering and extinction properties, shown in Sect. 4.
Single-particle DDSCAT calculations of the extinction coefficient
(Qext) for the dendrite aggregate (DA) and needle aggregate (NA)
at 50-micron effective radius. Shaded regions represent the range of
Qext due to various orientations of the particle relative to the
direction of incident radiation. Black lines are the equivalent ice sphere
Qext values at 1, 10, 50, and 100 % densities
(100 % is the bulk density of ice, 917 kgm-3).
(a) 13.4, (b) 35.6, (c) 94.0 GHz – note
different vertical axis ranges on each panel for optimal visualization.
Figure shows the single-particle DDSCAT calculations of the
extinction efficiency (Qext) for the DA and
NA at 50-micron effective radius for (a) 13.4, (b) 35.6,
and (c) 94.0 GHz. Shaded regions represent the range of variations in
Qext due to the range of orientations of the particle. Black
lines are the equal-mass ice sphere Qext values at 1, 10, 50, and
100 % densities (100 % is equivalent to a density of
917 kgm-3). The extinction increases rapidly with the onset of
melting. For every 0.05 increase in melting fraction, the extinction nearly
doubles at all three frequencies. Both the needle aggregate and dendrite
aggregate extinction efficiency exhibit roughly similar behavior, but it is
obviously different from the spherical properties and often outside of the
range that could be captured by spherical particle shapes alone
(e.g., ).
Same as Fig. 4 except effective radius is now 2500 microns (50-times-larger
radius; 125 000-times-larger mass). As before, the vertical axes are on
different scales. Notice the swap at 94 GHz, where the
needle aggregate exhibits a higher extinction than the dendrite aggregate.
Similar to Fig. 4, the single-particle backscattering efficiency coefficient
(Qbck) is shown for 50-micron effective radius. Both aggregates
show consistently larger backscattering efficiencies for the same mass as
melt fractions compared to the spheres. Note the difference in the vertical
axes on each panel.
Figure is the same as Fig. except that the effective
radius has increased to 2500 microns. The behavior at (a) 13.4 and (b) 35.6 GHz shows a similar sensitivity to the onset of melting, but not
quite as rapid as the smaller-particle case. Panel c (94 GHz) shows
a relative insensitivity of extinction to onset of melting; in this case it
is due to a trade-off between a rapidly decreasing scattering contribution and
equally increasing absorption contribution to the total extinction. Also of
note in panel c is that the needle aggregate now shows an overall larger
extinction than the dendrite aggregate, different from the other frequencies.
This is indicative of the increased sensitivity of the smaller wavelength
(approximately 3.2 mm at 94 GHz) to the finer-scale
structures present in the needle aggregate. We believe that this is the first
evidence in the existing literature that points towards this behavior;
however additional research will be required on this topic.
Same as Fig. 6 except at 2500-micron effective radius. The needle aggregate
at 35 GHz has a very large range due to particle rotation, whereas
the dendrite aggregate exhibits about half the same range. Similar to
Fig. 6c, the needle aggregate exhibits a larger backscattering compared to
the dendrite aggregate efficiencies at 35.6 and 94 GHz. Panel
(a) shows behavior consistent with spheres; however, in (b)
35.6 and (c) 94 GHz there is no consistent behavior. Note
that the backscattering efficiency in panel (b) for DA (blue
region) is completely encompassed by NA, and there is significant overlap
between the two in panel (c).
Extinction efficiencies (Qext) computed for both aggregates and
spheres at an effective radius of 1500 microns for (a) 89,
(b) 165, and (c) 183.31 GHz. Similar to previous
plots, shaded regions represent the ranges of Qext due various
orientations of the particle relative to incident radiation. Vertical axes
are on the same scale.
Following the same approach as Figs. and , Figs. and present the backscattering efficiency,
which is the differential cross section for backscattering (µm2 sr-1) divided by πaeff2 . Integrating the
single-particle backscattering cross section over a distribution of particle
sizes yields the radar reflectivity, which is discussed in
Sect. . Figure is the backscattering
efficiency for particles of a 50-micron effective radius, whereas
Fig. is the same at 2500-micron effective radius.
At 50-micron effective radius, the scattering is well into the Rayleigh
regime at all of the radar frequencies considered here. The response is
a relatively gentle increase in backscattering over the 0 to 0.15 melt
fraction range. The 94 GHz backscattering efficiency is roughly 3
orders of magnitude larger than the 13.4 GHz efficiency. Also of note
is that the variable density spheres (black lines) consistently
underestimates the total radar backscattering for all densities compared to
the two dendrites, implying that no modification of the density could
reproduce the backscattering obtained by the non-spherical particles.
Figure at 2500 microns shows a more complex relationship to
melting and particle orientation. The backscattering efficiency exhibits
a large variance due to particle orientation (the shaded regions) in all
panels. The spheres at 13.4 GHz encompass the range of variability
and exhibit a general increase in backscattering with melting. However this
breaks down at 35.6 and 94 GHz, where the Mie resonance effects start
to have a stronger influence on the computed backscattering. Similar to what
was seen in Fig. , the reversal of the backscattering roles occurs
at 35.6 and 94 GHz, with the needle aggregate exhibiting a higher
backscattering than the dendrite aggregate. It is also notable that at
94 GHz the backscattering decreases with increasing melt fraction.
Scattering and extinction at passive microwave frequencies
At passive microwave frequencies commonly employed for snowfall retrieval
(e.g., 89, 165, and 183.31 GHz), the particle interaction with
microwave radiation comes primarily through extinction (scattering + absorption) and emission. Due to space constraints, only one effective radius
(1500 microns) is shown here in order to understand the general sensitivity
of the single-particle extinction, single-scattering albedo and asymmetry
parameter to the onset of melting.
Similar to Fig. 8 except plotting single-scattering albedo (ω̃=Qsca/Qext). See text for discussion.
In Fig. we have computed the extinction efficiency for the
NA and DA vs. melt fraction. Generally the
spheres do not adequately cover the observed ranges of extinction for the
non-spherical particles. At 89 GHz (panel a), the DA extinction is
higher than NA; this role reverses at 165 and 183.31 GHz. Roughly
speaking, 165 and 183.31 GHz exhibit similar sensitivities to melting
at all effective radii (only 1500-micron effective radius is shown here), suggesting that, for
rough estimates, computing the 165 GHz properties may be sufficient
for capturing the scattering and extinction behaviors at 183.31 GHz
(and nearby channel offsets employed on the GPM microwave imager, and other
imagers and sounders). In actual remote-sensing applications, the difference
in water vapor emission/absorption at 165 vs. 183.31 GHz is likely to
dominate the signal, except in the driest of atmospheric
profiles .
The single-scattering albedo (the ratio of scattering to the total
extinction) presented in Fig. tells us the primary story of
interest with regard to the onset of melting. Specifically we see that
a small change in melt fraction yields a significant linear decrease in
single-scattering albedo, which is an indicator of the rapidly increasing
contribution of absorption to the total extinction. This, in turn, drives the
thermal emission (according to Kirchhoff's law) that we will later observe in
Sect. .
Following Figs. 8 and 9, the scattering asymmetry parameter (indicating the
degree of forward scattering) is shown here. At all frequencies, the
asymmetry parameter is relatively insensitive to the early stages of melting,
consistent with the fact that the overall structure of the melting particles
does not change too much over this range. In panel (a), the
asymmetry for NA (blue region) is fully encompassed by the asymmetry from
DA, with partial overlaps in (b, c).
Finally, in Fig. , the scattering asymmetry parameter
(cosine-averaged scattering contribution over all angles) describes the degree to
which incident radiation is forward scattered (g>0) or backward scattered
(g<0). At a 1500-micron effective radius, very little sensitivity to the
onset of melting was observed. Similar results were found at other effective
radii. This is consistent with the notion that the scattered radiation
depends primarily on the shape of the particle. Over this range of melting,
the actual particle shape has changed very little, so the degree of forward
scattering is not expected to change much. If melting were to continue beyond
a melt fraction of 0.15 (not shown), the asymmetry parameter would change as
the shape of the particle changes.
Overall, the single-scattering properties show a marked sensitivity to the
onset of melting for scattering and extinction, with the exception of the
asymmetry parameter. The spherical particle approximation does not produce
scattering and extinction properties that have similar behavior to the
non-spherical particle, particularly as the frequencies change. In some
cases, the spherical particle properties do not bracket the non-spherical
particles' properties, suggesting that under the current formulation no
amount of modifying the density parameter could result in a reliable
substitute for more physically realistic shapes when one considers all of the
scattering and extinction properties of interest to passive and active remote-sensing applications.
Simulated radar reflectivities at (a) 13.4, (b) 35.6, and
(c) 94 GHz. Reflectivities were computed by integrating over
the orientation-averaged single-particle backscattering cross sections,
assuming an exponential particle size distribution, for both aggregates and
spheres. The reflectivities are presented only for one value of
D0=0.11cm (i.e., the liquid-equivalent median volume diameter).
N0 and D0 are computed following (see text). Note
different scaling on the vertical axes.
Integrated properties
In the previous section, we examined a subset of the
single-particle scattering and extinction properties. Of interest for
atmosphere radiative transfer and remote-sensing applications is how these
quantities behave in an ensemble of particles.
Radar response
The equivalent radar reflectivities (Ze) at each frequency are
calculated using the single-particle radar backscattering cross sections,
Cbck(D), and integrating these over a given particle size
distribution, N(D):
Ze=λ4π5|Kw|2∫DminDmaxCbck(D)N(D)dD,
where D is the mass-equivalent diameter (twice the effective radius),
λ is the wavelength of incident radiation with the same units as D,
and Kw=(mw2-1)/(mw2+2) is the
dielectric factor computed from the refractive index of water,
mw. We note that in the present case we specifically chose
|Kw|2≈0.93 for all wavelengths and melt fractions as a
comparative convenience, so that only the backscattering properties vary with
melt fraction. In actual radar reflectivities, the response to the dielectric
properties of the ice/snow/mixed-phase particles within the radar range
gate will be different, and completely dependent on the individual
constituents shape, composition, temperature, and frequency of incident
radiation.
For lack of a suitable alternative, we have assumed that the melt fraction of
each size of particle is the same fraction across all particles in the
distribution. This provides a constant melt fraction quantity, independent of
mass, so that radar sensitivity to variations in the melt fraction can be
readily examined without confusion. Future research will explore the
variation of melt fraction for particles of different sizes in a given volume
of the atmosphere.
For the snowfall particle size distribution, we choose the exponential size
distribution from and . In each of the following calculations,
all comparisons are made using an equal particle mass distribution. The
particle size distribution N(D) is given by
N(D)=N0exp-3.67DD0,
where N0 is the intercept parameter and D0 is the “characteristic” diameter of the PSD.
N0 and D0 are related as follows:
D0=0.14R0.45[cm],N0=2.5×103R-0.94[mm-1m-3],
when liquid-equivalent precipitation rate, R, is given in units of millimeters per hour (mm hr-1).
Figures and show the simulated equivalent radar
reflectivity, averaged over all particle orientations, vs. melt fraction for
the NA (green line) and DA (blue
line), along with the mass-equivalent variable-density spheres (black lines).
In Fig. 11, we have selected the characteristic diameter D0=0.11cm (corresponding to an ice water content of approximately
0.13 gm-3) for analysis. At all three frequencies,
the reflectivity increases by a few decibels over the range of melting. With
a judicious choice of density, spheres could reasonably approximate the
simulated reflectivities of the non-spherical particles, except for the
dendrite aggregate at 13.4 GHz (Fig. a).
Similar to Fig. 11 except for a larger D0=0.55cm. Notice the
flip in the computed reflectivities at 94 GHz, similar to what was
observed in the single-particle backscattering properties (Fig. 7). Note: the
results in this figure may be subject to large particle diameter truncation,
as discussed in the text.
Volume extinction coefficient for D0=0.11cm at (a)
13.4, (b) 35.6, and (c) 94.0 GHz. The
orientation-averaged single-particle extinction cross section was integrated
over the same particle size distribution as was used in Fig. 11. Note
different scaling on the vertical axes, including the exponents.
In Fig. , the orientation-averaged reflectivities are computed
for D0=0.55cm, corresponding to an ice water content of
approximately 3 gm-3, which is roughly
15 mmh-1 liquid-equivalent snowfall
rate – an exceptionally high snowfall rate with very large snowflakes,
representing an upper limit. Consequently, the simulated equivalent radar
reflectivities are as high as 47 dBZ for DA at 13.4 GHz. The
increase in reflectivity over the onset of melting is, similar to
Fig. , only a few decibels per Z (dBZ) from 0 to 0.15 melt fraction.
We note that because D0=0.55cm is greater than the upper limit of the individual
particle size, truncation errors in the particle size distribution integration are potentially significant.
So these quantities should be interpreted with this in mind. Due to computational
limitations at the time of research, the individual melt particle sizes cannot be reliably
computed above 2500-micron effective radius (5000-micron effective diameter).
Although not shown here, continued melting does not significantly increase
the reflectivity beyond this point. This result may appear to be inconsistent
with radar observations of melting precipitation (i.e., the radar
bright band) . This suggests that the changing dielectric properties
alone do not create the observed rapid increase in radar reflectivities
associated with melting. It is postulated, without proof, that enhanced
aggregation of particles at or near the melting point causes a rapid increase
in total particle size, enhancing the D4 diameter dependence of the
reflectivity factor (e.g., ). A more detailed analysis of
enhanced aggregation is beyond the scope of the current research.
Same as Fig. 13 except for D0=0.55cm. Note the different
scaling on the vertical axes, including the exponents; the results in
this figure may be subject to large particle diameter truncation, as
discussed in the text.
A surface plot of simulated brightness temperatures (TBs) for D0 vs. melt
fraction assuming an “infinite” layer of melting hydrometeors at
(a) 89, (b) 165.0, and (c) 183.31 GHz.
A two-stream radiative transfer model (see text) was used to compute the TBs.
Particle size distributions used here are the same as those used in the
reflectivity computations.
Figures and are physically consistent with
Figs. and , respectively, but now show the volume
extinction coefficient. The most apparent feature is the rapid increase in
extinction with the onset of melting. This behavior was observed in the
single particle scattering properties (e.g., Figs. –).
The integrated extinction suggests that attenuation of the radar beam starts
to accumulate rapidly with only a modest increase in reflectivity. As before,
the range of extinction of the spherical particle approximation does not
always encompass the extinction of the two selected non-spherical particles
in Fig. a and c.
Passive microwave response
In addition to the radar sensitivity to melting, there is also an interest in
understanding the sensitivity of passive microwave brightness temperatures to
the onset of melting. In reality, the melting layer of a precipitating cloud
is likely to be obscured by an overlying ice region, obscuring it partially
or wholly, depending on the wavelength of radiation. For the following
simplified analysis, a single layer of melting hydrometeors is simulated,
with no atmospheric gases or other layers intervening. To compute brightness
temperatures, a two-stream approximation is used, which in past studies
has been found to be remarkably accurate at describing
thermal emission from a plane-parallel slab viewed at the 55∘
incidence angle, typical of a satellite microwave imager. The details of the
two-stream approximation are provided in . Under the assumption
of an infinitely thick slab, the transmittance through the slab is zero.
Consequently the resulting upwelling brightness temperature from the
two-stream approximation is only a function of the layer temperature, Ta,
and the layer reflectivity, r∞:
TB,∞=Ta-Tar∞,
where
r∞=1-gω̃-1-g1-gω̃+1-g.
Here ω̃ and g have been appropriately integrated over the
exponential particle size distribution (Eq. ). In this simplified
model, r∞ is uniquely determined given the following physical
properties: the wavelength of incident radiation, melting fraction (i.e., the
dielectric properties), the particle orientation, and the particle size
distribution. The utility of the two-stream model should not be oversold: it
is a useful method for understanding the bulk sensitivity of upwelling
microwave TBs to modifications in the underlying physical properties of
hydrometeors, but it ignores all other contributions that would normally be
present in an actual remote-sensing scene. In this sense, it provides a
“worst-case” scenario for the influence of microphysical properties on the
upwelling brightness temperature.
Figure shows a surface plot of two-stream brightness
temperatures computed at 89, 165, and 183.31 GHz. The y axis is
D0, and the x axis is the melt fraction from 0 to 0.15. Given a fixed
D0 (i.e., a fixed ice water content), the onset of melting can have
a rapid and dramatic impact on the simulated TBs. For example, with D0 around
0.03 cm, proceeding from an unmelted particle to a melt fraction of
0.01 (only 1 % melted), the brightness temperatures increase by
almost 100 K, whereas for larger particles (higher ice water
contents) the sensitivity to melting is not as rapid but nevertheless
significant throughout the range of D0 values. The onset of melting, from
0.0 to 0.15 melt fractions, leads to increases of up to approximately
125 K. In order to better understand the impact of this, further
analysis is needed wherein a 1-D vertical or 3-D simulation is used to
simulate a more realistic atmospheric column. It is expected that an ice
layer overlaying the melting layer would partially or wholly obscure
emission through scattering at the frequencies examined here – this is left
for future studies.
Conclusions
In this paper, the Single Particle Melting Model was introduced as
a novel and computationally efficient method to simulate the melting of an
arbitrarily shaped ice hydrometeor. SPMM uses a novel nearest-neighbor method
for determining when a particular point will melt and when previously melted
points can move. It is easy to implement and map into any existing
thermodynamic/melting-layer model, where a simple mapping between meltwater
generation in a thermodynamic model and the melt fractions generated by the
melting simulation are linked as appropriate. This also provides finer
control over the particle size distribution within the melting layer.
A limited study of the onset of melting, for melt fractions ranging from 0 to
0.15, was performed in order to quantify the sensitivity of microwave
radiation scattering and extinction. Two snowflake aggregate shapes were
selected: one composed of needles and the other composed of dendritic
crystals. For comparison with past studies, the scattering and extinction
properties of spherical particles having 1, 10, 50, and 100 % volume
fractions of ice (relative to air) were simulated. Single particle
calculations using the discrete dipole approximation, via the DDSCAT code,
were made, highlighting the individual particle and size-distribution
integrated particle properties. These calculations were made at 13.4, 35.6,
and 94 GHz, consistent with the GPM dual-frequency precipitation
radar, and at 89, 165, and 183.31 GHz, consistent with the GPM
microwave radiometer. We believe that this study represents the first
simulation of the scattering and extinction properties of
realistically shaped melting hydrometeors for a wide range of microwave
frequencies and particle sizes.
There is a significant sensitivity of the computed extinction and scattering
properties to the base hydrometeor shape and to the onset of melting. We
found, in particular, that the spherical particle assumption was unable to
capture the range of computed scattering properties from the non-spherical
particles and did not provide consistent relationships between scattering
and extinction throughout the onset of melting. The conclusion one could draw
from this is that from a modeling perspective it appears that spherical
particles (no matter how the density/mass is modified) cannot fully represent
the range of uncertainties in the absence of of knowledge of the hydrometeors present
in a given remote-sensing scene. Capturing this behavior in physical models
is critical for accurately computing uncertainty estimates in forward model
simulations and retrieval algorithms.
Validation of these simulations could be, in future work, performed by
examining, for example, radar observations of stratiform melting layers –
especially in cases where in situ observations of particle shape are
available. The present model is currently being adapted to simulate melting
for existing ice hydrometeor databases consisting of tens of thousands of
particles, and it will allow for more realistic comparisons with observational
data.