A simplified approach is presented for assessing the microwave response to
the initial melting of realistically shaped ice particles. This paper is
divided into two parts: (1) a description of the Single Particle Melting
Model (SPMM), a heuristic melting simulation for ice-phase precipitation
particles of any shape or size (SPMM is applied to two simulated aggregate
snow particles, simulating melting up to 0.15 melt fraction by mass), and (2)
the computation of the single-particle microwave scattering and extinction
properties of these hydrometeors, using the discrete dipole approximation (via
DDSCAT), at the following selected frequencies: 13.4, 35.6, and 94.0

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.,

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
(

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

In this study, the discrete dipole approximation (DDA), using DDSCAT (Discrete Dipole Scattering) version
7.3

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

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

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.

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

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.

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.

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

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

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.

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

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,

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.

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

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

“High-frequency” passive microwave channels at 89, 165, and
183.31

Overview of 13.4

In Fig.

Single-particle DDSCAT calculations of the extinction coefficient
(

Figure

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

Similar to Fig. 4, the single-particle backscattering efficiency coefficient
(

Figure

Same as Fig. 6 except at 2500-micron effective radius. The needle aggregate
at 35

Extinction efficiencies (

Following the same approach as Figs.

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

Figure

At passive microwave frequencies commonly employed for snowfall retrieval
(e.g., 89, 165, and 183.31

Similar to Fig. 8 except plotting single-scattering albedo (

In Fig.

The single-scattering albedo (the ratio of scattering to the total
extinction) presented in Fig.

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

Finally, in Fig.

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

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.

The equivalent radar reflectivities (

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

Figures

Similar to Fig. 11 except for a larger

Volume extinction coefficient for

In Fig.

We note that because

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)

Same as Fig. 13 except for

A surface plot of simulated brightness temperatures (TBs) for

Figures

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

Here

Figure

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

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.

The majority of this investigation was supported by the NASA PMM (R. Kakar) and RST (L. Tsaoussi) programs – specifically NASA grants NNX11AR55G (PI: B. Johnson), NNX10AI49G (PI: W. Olson), and NNX10AT36A (PI: G. Skofronick-Jackson). Edited by: M. Kulie