Drop-scale numerical modeling of chemical partitioning during cloud hydrometeor freezing

Full text

DROP-SCALE NUMERICAL MODELING OF CHEMICAL PARTITIONING
DURING CLOUD HYDROMETEOR FREEZING
A.L. Stuart1 and M.Z. Jacobsonr2
1Department of Atmospheric Sciences, Texas A&M University, College Station, TX, 77843, USA
2Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, 94305, USA
1. INTRODUCTION
Thunderstorms can significantly impact chemical
distributions in the troposphere by 1) redistribution of
air and hydrometeors containing trace chemicals and
2) providing a multi-phase environment for chemical
phase changes and reactions. Interactions between
ice-containing hydrometeors and chemicals are not
well understood. Laboratory and field measurements
of chemical partitioning during drop freezing provide
greatly varying estimates of the retention efficiency of
volatile solutes (e.g., Lamb and Blumenstein, 1987;
Iribarne and Pyshnov, 1990, Snider and Huang, 1998;
Voisin and Legrand, 2000). Our recent work using a
theory based time scales analysis to calculate the
retention efficiency provided a basic understanding of
the dependence of partitioning on chemical properties
and freezing conditions (Stuart and Jacobson, 2003,
2004). In this work, we develop a drop-scale time-
dependent numerical model of drop freezing and
chemical transfer to investigate the effects on
chemical partitioning of dynamical interactions
between involved processes.
2. MODEL DESCRIPTION
Our one-dimensional (radial) model represents the
freezing of a spherical solute-containing liquid
hydrometeor with a riming substrate or ice nuclei at its
center. Processes represented by the model include
radial inter- and intra-phase heat and mass transfer,
freezing kinetics, latent heat release during freezing,
and solute segregation and trapping at the freezing
interface.
2.1 Process Representation
The model is initiated at freezing nucleation. As time
progresses, the drop freezes and solute is expelled at
rates governed by both the grid-resolved partial
differential equations for radial heat and mass transfer
as well as the sub-grid scale differential equations for
processes occurring near the ice water interface. We
represent the multi-phase (dendritic) character of
freezing in the super-cooled drop using average water
and ice amounts (fractions) in each radial shell
_________________________________________
Corresponding author’s address: Dr. Amy L. Stuart,
Dept. of Atmospheric Sciences, Texas A&M
University, 3150 TAMU, College Station, TX 77843-
3150; E-Mail: amystuart@tamu.edu.
volume. Properties and prognostic variables
(temperatures and solute concentrations) are also
volume average values for each phase. This ‘mushy
zone’ representation (after Tien and Geiger, 1967) of
the non-planar ice-water interface assumes phase
changes (freezing) provide a volume source of latent
heat. The density of water and ice is assumed to be
constant and equivalent in both phases.
Processes resolved on the model grid include radial
intra- and inter-phase heat and mass transfer.
Equations for diffusive transfer for each phase were
derived using energy and mass balances over a shell.
Inter-phase transfer coefficients were based on two-
film theory (e.g. Sherwood et al., 1975) and account
for mass transfer resistances in both phases.
Boundary conditions at the grid’s outer boundary (the
hydrometeor surface) account for flux due to
conduction, evaporation, sublimation, and
condensation. Flux enhancement due to fluid flow
around the hydrometeor is represented with a flow-
dependent ventilation coefficient.
Processes represented at the sub-grid scale include
phase change, latent heat transfer, inter-phase heat
transfer, and solute mass segregation during phase
change. Rates of freezing (and melting) are
represented by a kinetic equation for the interfacial
growth velocity, which is based on theory and
experimental data on growth rates of ice in super-
cooled water (Bolling and Tiller, 1961; Pruppacher and
Klett, 1997). We estimate the interfacial surface area
as the ratio of the shell surface area to its volume. To
simulate dendrite-like growth, ice initiation in any cell
can only occur if an adjacent shell contains ice. To
represent latent heat transfer, we perform an enthalpy
balance over the grid shell. Energy distribution to
each phase is weighted by the volume fraction of each
phase in a shell. For phase initiation in a given shell,
we assume the temperature in both phases remains
equal. Inter-phase heat flux is represented as
conductive transfer for which the bulk heat transfer
coefficient is a function of the two-film inter-phase
transfer coefficient and the dendrite tip radius. We
assume free dendritic crystal growth and calculate the
dendrite tip radius as a function of the dendrite Peclet
number and the interfacial growth velocity (after Caroli
and Muller-Krumbhaar, 1995, and Libbrecht and
Tanusheva, 1999). Solute mass segregation during
freezing is calculated to maintain a mass balance,
assuming equilibrium partitioning at the ice-water

interface. During melting, all mass in the melted ice is
assumed to be transferred to the water phase.
Trapping of concentrated solute between dendrite
branches, leading to bulk non-equilibrium partitioning
to ice, is represented by allowing grid shells that
freeze completely in a given time step to retain all
solute originally in that shell.
2.2 Flow of calculations and numerics
The flow diagram for model calculations is provided in
Figure 1. The model is initiated by specifying ambient
conditions (constant temperature and pressure),
hydrometeor conditions (initial drop and ice substrate
sizes, temperatures, and speed in air), initial solute
concentrations in all phases, the outer-model time-
step (used for grid-resolved processes), and the
number of radial grid shells. The grid is determined
and initialized with these inputs, assuming the drop
water is spread evenly around the ice substrate.
Following initialization, the model time cycle of
processes begins. For each outer-model time-step,
sub-grid processes of phase change with latent heat
transfer, inter-phase heat transfer, and solute
segregation and trapping are first calculated. Each
process is calculated separately in a serial manner
(i.e. processes are time-split). Phase change with
latent heat transfer and inter-phase heat transfer
calculations use an adaptive time-step to ensure
physically-consistent results (e.g., temperatures at the
interface that do no significantly overshoot the
equilibrium freezing temperature). The grid-resolved
processes of radial heat and mass transfer are then
calculated for the outer-model time-step. A finite
volume 2nd-order central difference discretization is
used for calculation of radial fluxes. Progression in
time is discretized with a Forward Euler formulation.
After each outer-model time-step, values of system
enthalpy and mass (of water and solute) are
calculated to track conservation properties of the
simulation. The model is terminated when the
hydrometeor is completely frozen.
3. MODEL DEMONSTRATION
To test and demonstrate the model, we have applied it
to simulate several cases of freezing of drops of
varied sizes and ambient conditions. We will detail
results for one demonstration case here and then
briefly discuss model performance for all cases.
3.1 Case Description
Our demonstration case simulated the freezing of a
hydrometeor falling at its terminal fall speed, formed
due to the impaction of a supercooled drop (1000 mm
in radius) with a ice substrate (100 mm in radius). The
air and initial supercooled drop temperatures were
–10°C. The initial ice substrate temperature was
–5°C. The ambient pressure was 300 mbar. For
demonstration, we used a hypothetical chemical
Initialization
Mass and enthalpy
conservation
Output
Freezing / melting and
latent heat transfer
Inter-phase heat
transfer (sub-grid)
Solute segregation
and trapping
Radial heat and
mass transfer
n = n + 1
completely
frozen?
End
Yes
No
Figure 1. Flow diagram of model calculations, where n is the
outer-model time-step.
solute with the following properties: dimensionless
(concentration) Henry’s constant of approximately 30,
ice-water partition coefficient of 0, and diffusivities in
air, water, and ice of 0.1 cm2/s, 1x10-5 cm2/s, and
1x10-10 cm2/s, respectively. Solute concentrations in
the gas phase and supercooled drop where initially at
equilibrium, with values of 7x10-7 g/cm3 and 2x10-5
g/cm3, respectively. Initial solute concentration in the
ice substrate was 0. The number of radial grid shells
was specified as 10 and the outer-model time-step
was 1x10-4 s.
3.2 Case Results
We will first describe the freezing progression and
temperature changes in the hydrometeor. The
hydrometeor is initially liquid throughout most of its
radius, with a solid core (the original ice substrate).
Ice quickly propagates out from the ice core, reaching
the hydrometeor surface by 9x10-4 s. Ice fraction
values throughout the mixed-phase region remain
below 0.1 during this time. Water and ice
temperatures in the mixed-phase region also increase
from the inside of the hydrometeor outward (due to
release of latent heat of freezing). Temperatures
reach an approximately uniform value slightly below
273 K by 2x10-3 s. Once the mixed-phase region
temperatures are approximately equivalent, the
temperatures and ice fractions increase more slowly
and uniformly. By approximately 4x10-3 s, we see the
ice fraction near the air boundary surpass that in the
hydrometeor interior. The temperature in the solid
core (the original ice substrate) increases slowly (due
to radial heat transfer from the mixed-phase zone), but
does not reach the temperature of that zone until
about 3x10-2 s. Far from the air boundary, the
temperatures reach the equilibrium freezing
temperature of water at approximately 0.1 s.

Temperatures near the air boundary are depressed
(due to heat loss to air that is 263 K). By 7 seconds,
an ice-shell (ice fraction of 1.0) has formed at the
hydrometeor surface. Freezing propagates inward
until the entire hydrometeor is frozen by 24.6 s.
Temperatures (of ice) also decrease from the outside
inward after the freezing front.
This progression of freezing and temperature change
can be compared with experimental and theoretical
studies of drop freezing (e.g. see Macklin and Payne,
1967, 1968; Griggs and Choularton, 1983;
Pruppacher and Klett, 1997). These studies suggest
that nucleated drops freeze in approximately two
stages. The first stage is termed the adiabatic stage.
During this stage, ice propagates out from the
nucleation site and the drop heats up to the
equilibrium freezing temperature of water, with
relatively little heat loss to the drop environment.
(This is termed the adiabatic stage.) This stage is
very quick, orders of magnitude faster than the
complete freezing of the drop. The second stage is
termed the diabatic stage. During this stage the
freezing occurs more slowly, limited by the rate of heat
loss to the ice substrate and the surrounding air. This
picture of freezing progression is qualitatively
consistent with our results.
For quantitative comparison, we estimate the
adiabatic freezing stage during our simulation to be
within an order of magnitude of 0.01 s. (Bounded by
the time it takes for ice to reach the drop surface,
approximately 0.001 s, and the time it takes the
mixed-phase zone to heat to approximately the
equilibrium freezing temperature, 0.1 s). During this
time, approximately 13% of the drop mass froze in our
simulation. This is in excellent agreement to the
value, cwDT/Lm (or 13%), derived from drop freezing
theory (Pruppacher and Klett, 1997). The complete
drop freezing time from our simulation, 24.6 s, can
also be compared to the bulk theoretical expression
developed for drops falling freely in air (Pruppacher
and Klett, 1997). This expression yields a value of
19.4 s for our simulated conditions, which is about
21% lower than our simulated value.
The progression of solute redistribution during
freezing was also simulated. Solute concentrations in
liquid water were initially 2.0x10-5 g/cm3 throughout
the hydrometeor and 0 in the ice core. As ice
propagates outward, solute concentrations in water
away from the air boundary increase slightly to
2.3x10-5 g/cm3 at 1 s, due to the exclusion of solute
from the ice phase during freezing. Concentrations
near the air boundary decrease to 1.9x10-5 g/cm3 at 1
s, due to mass transfer to air. After the outer shell of
the hydrometeor freezes, and freezing propagates
inward, concentrations in water near the inner
boundary of the shell increase more dramatically to
about 1x10-4 g/cm3 due to exclusion from the ice
phase and lack of a sink to air. In ice, average
concentrations within and near the original ice core
jump to 7.0x10-9 g/cm3 directly after freezing initiation,
due to freeze trapping. As ice propagates through the
hydrometeor, solute concentrations in ice increase to
non-zero values, due to radial mass transfer. When
the outer shell of the hydrometeor freezes, the
concentration in ice near the air boundary jumps to
9.4x10-8 g/cm3, also due to freeze trapping. As
freezing progresses inward, higher concentrations are
trapped in ice due to higher concentrations in the
liquid. The final solute concentration profile indicates
increasing solute concentrations inward, with the area
near the hydrometeor surface (air boundary) having
very low concentrations (approximately 1x10-7 g/cm3)
and the area frozen last (near the original ice core)
having a concentration slightly higher that that
originally in water (2.3x10-5 g/cm3). The total retention
efficiency in the hydrometeor (the ratio of the mass of
solute in the hydrometeor to that originally in the drop)
decreases precipitously from 1.0 to 0.96 in the first 0.1
s of freezing. It continues to decrease more slowly
until an ice shell forms at the surface (at about 7 s).
After ice shell formation, there is negligible loss and
the retention fraction remains constant at 0.72. These
results are qualitatively physical and consistent with
the body of literature on crystallization separation (e.g.
Zief and Wilcox, 1967). However, they cannot be
quantitatively compared to experimental results due to
our use of a hypothetical solute for demonstration.
3.3 Model Performance
For all cases simulated, freezing and solute
redistribution occurred in a similar manner.
Quantitative comparison of simulated freezing times to
theoretical bulk estimates of freezing times yielded
values in reasonable agreement (differences within
approximately 20%). Mass and heat conservation was
excellent for all case simulations, with exact solute and
water mass balance and a very small (insignificant)
heat balance error.
4. CONCLUSIONS
We have developed and demonstrated a one-
dimensional radial model of chemical partitioning
during hydrometeor freezing. Model results are
physically consistent with freezing, heat, and mass
transfer theory. The model is exactly mass conserving
and also shows good energy conservation. Insights
provided by results from model demonstration cases
include that the location of nucleation within the drop
may not be very important to overall drop freezing.
Due to fast dendritic growth, the freezing front quickly
moves to the drop surface and overall freezing
progresses inward from the surface. Trapping due to
ice shell formation at the hydrometeor surface may
also play a large role in non-equilibrium bulk solute
partitioning to the ice phase hydrometeor. Further
development of the model will include 1) improvement
of the dendrite initiation representation to account for
the distance of tip travel, 2) improvement of the solute
trapping representation through use of an effective ice-

liquid distribution coefficient dependent on the
freezing rate and/or dendrite characteristics, 3)
detailed testing of the freezing and chemical mass
transfer modules, and 4) addition of a chemical
reaction solver. We plan to apply the model to
investigate the dynamics of chemical partitioning and
reaction during freezing and riming in clouds. Results
from these investigations will inform modeling and
understanding of chemical scavenging and
redistribution by clouds.
5. REFERENCES
Bolling, G. F. and W. A. Tiller, 1961: Growth from the
Melt. III. Dendritic Growth. Journal of Applied Physics,
32, 2587-2605.
Caroli, B. and H. Muller-Krumbhaar, 1995: Recent
advances in the theory of free dendritic growth. ISIJ
International, 35, 1541-1550.
Griggs, D. J. and T. W. Choularton, 1983: Freezing
modes of riming droplets with application to ice
splinter production. Quarterly Journal of the Royal
Meteorological Society, 109, 243-253.
Iribarne, J. V. and T. Pyshnov, 1990: The effect of
freezing on the composition of supercooled droplets -
I. Retention of HCl, HNO3, NH3, and H2O2.
Atmospheric Environment, 24A, 383-387.
Lamb, D. and R. Blumenstein, 1987: Measurement of
the entrapment of sulfur dioxide by rime ice.
Atmospheric Environment, 21, 1765-1772.
Libbrecht, K. G. and V. M. Tanusheva, 1999: Cloud
chambers and crystal growth: effects of electrically
enhanced diffusion on dendrite formation from neutral
molecules. Physical Review E, 59, 3253-3261.
Macklin, W. C. and G. S. Payne, 1967: A theoretical
study of the ice accretion process. Quarterly Journal
of the Royal Meteorological Society, 93, 195-213.
Macklin, W. C. and G. S. Payne, 1968: Some aspects
of the accretion process. Quarterly Journal of the
Royal Meteorological Society, 94, 167-175.
Pruppacher, H. R. and J. D. Klett, 1997: Microphysics
of Clouds and Precipitation. 2nd ed. Vol. 18, Kluwer
Academic Publishers, 954 pp.
Sherwood, T. K., R. L. Pigford, and C. R. Wilke, 1975:
Mass Transfer. McGraw-Hill Book Company.
Snider, J. R. and J. Huang, 1998: Factors influencing
the retention of hydrogen peroxide and molecular
oxygen in rime ice. Journal of Geophysical Research,
103, 1405-1415.
Stuart, A. L. and M. Z. Jacobson, 2003: A timescale
investigation of volatile chemical retention during
hydrometeor freezing: Nonrime freezing and dry
growth riming without spreading. Journal of
Geophysical Research, 108, 4178,
doi:10.1029/2001JD001408.
Stuart, A. L. and M. Z. Jacobson, 2004: Chemical
retention during dry growth riming. Accepted for
publication in Journal of Geophysical Research.
Tien, R. H. and G. E. Geiger, 1967: A heat-transfer
analysis of the solidification of a binary eutectic
system. Journal of Heat Transfer, 89, 230-234.
Voisin, D. and M. Legrand, 2000: Scavenging of acidic
gases (HCOOH, CH3COOH, HNO3, HCl, and SO2)
and ammonia in mixed liquid-solid water clouds at the
Puy de Dome mountain (France). Journal of
Geophysical Research, 105, 6817-6835.
Zief, M. and W. R. Wilcox, 1967: Fractional
Solidification. Vol. 1, Marcel Dekker, Inc, 714 pp.