the wire, the rate of precipitation, the wind speed,
reviews relationships between precipitation rate
and the rate at which the impinging water freezes.
and liquid water content that both he and other
The models, however, do not attempt to deter-
researchers, including Marshall and Palmer (1948),
mine the ice accretion shape, but instead assume
had determined. I chose his formula rather than
the MarshallPalmer formula (W = 0.072P0.88) for
simple shapes that, I hope, adequately represent
reality.
ZRAIN because it is the average of all the W(P)
In the simple flux model, the ice accretes with a
relationships, while MarshallPalmer's formula
uniform radial thickness around the circumfer-
gives comparatively high values of W. In ZRAIN,
ence of the wire. This simple shape is consistent
the raindrops move horizontally at the wind speed,
with the level of detail in the model.
so the flux of precipitation w is the vector sum of
the vertical and horizontal water fluxes, Pρo and
In the heat-balance model, the precipitation
that freezes immediately accretes with a uniform
WV, respectively, converted to consistent units:
radial thickness. The runoff water that does not
1/ 2
Pρ 2
freeze immediately is allowed to freeze as icicles.
2
2
w (g/m s) =
+ (WV)
o
.
(3)
3.6
The "icicle" in this model is a single, vertically
oriented, circular cylinder of ice per meter of wire
that can be understood as the actual icicles stacked
Nine heat-flux terms were considered for inclu-
up end to end. As runoff water freezes to the sides
sion in the heat-balance model to determine the
of the icicle, its diameter increases from an initial
fraction of the available precipitation that freezes.
assumed diameter of 0.5 cm (Maeno et al. 1994).
They are:
Runoff water that does not freeze to the sides of the
icicle is assumed to flow down to its tip where it
Qw (cooling flux required to warm
the incoming water to 0C) =
may freeze, in successive 0.5 cm-diameter half
spheres, and increase the length of the icicle. Water
cww(0 T)
that does not freeze there is assumed to drip off. At
Qc (convective cooling flux) =
πh(0 T)
the end of each time step, the total icicle mass is
distributed evenly over the new icicle length to
Qe (evaporative cooling flux) =
102πhmLe[e0/273.15 RheT/(T
form a uniform circular cylinder. For each 10-min
time step the diameter of the wireice accretion Dc,
+ 273.15)]mw/R
and the diameter Di and length Li of the icicle are
Ql (longwave radiative cooling flux)
= πσ[273.154 (T + 273.15)4]
determined as initial conditions for the next time
step.
πSr/2
(4)
Qf (flux of heat released by freezing
all impinging precipitation) = Lf w
4. HEAT BALANCE IN FREEZING RAIN
103πhrvV2/(2cp)
The primary process in determining the amount
of the available precipitation that freezes to a
Qk (heat flux from droplet kinetic
structure is the flux of heat from the accretion
energy) =
103w{V2 + [Pρo/(3.6W)]2}/2
surface. Heat must be removed from the rain drops
that collide with a structure for them to freeze. In
calculating the heat fluxes in the heat-balance
rent flowing through a conductor) =
102I2 Rc/Dc.
model, the wireice accretion is assumed to be at
0C. This is usually a good assumption in freezing
rain because, in the typical near-freezing air tem-
The heat fluxes are in watts per unit length of the
peratures, relatively low winds, and high water
wire per wireice diameter. Qc, Qe, Ql, Qs, and Qv
all include a factor of π because they are for the
flux, not all the impinging precipitation freezes.
The flux of precipitation is computed using the
outside surface of the wireice accretion, which
has an area per unit length of π times the diameter.
relationship from Best (1949) to determine the
liquid water content of the air as a function of the
precipitation rate,
compared in Figure 3 for ranges of temperatures,
wind speeds, precipitation rates, relative humid-
W = 0.067P0.846,
(2)
ties, and diffuse solar radiation typical for freez-
ing-rain events. The individual heat-flux terms are
where W is in g/m3 and P is in mm/hr. Best
described below.
6
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