The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 38

ticles and hydrometeors everywhere simultaneously in a convective storm for its
entire duration is a major obstacle in our quest to understand completely and to
be able to predict the evolution of convective storms. It will be seen later that
polarimetric radars offer some hope for bridging the gap in our being able to map
out hydrometeors three dimensionally in a convective storm.
It is not practical to follow individual hydrometeors around in a convective
storm; the spatial resolution required is much greater than the spatial resolution
used in numerical cloud models. Microphysical processes are therefore parameter-
ized, just as subgrid-scale turbulence is parameterized based on ''bulk'' quantities
(i.e., those on larger, resolvable scales). The most accurate technique for represent-
ing the microphysical processes in (2.46) in numerical models is the ''bin'' (also
known as ''spectral'') method. In the bin method, the size or mass of hydrome-
teors within each bin (i.e., range of diameter or mass) is apportioned according to
microphysical and kinematic processes. When many bins are used, as is necessary
for realistic results, a relatively long duration of computer time is needed.
To reduce computer time, bulk multi-moment parameterization schemes are
often used. In this case, a specified distribution function for hydrometeor size
(assumed a priori and not generated for each case, as happens with the bin
method) is used. For a quantity M

ð 1

D k N ð D Þ dD

M k ¼ ð 2 : 47 Þ

0

where k is the moment; D is the diameter of the hydrometeor; N ð D Þ is the
hydrometeor size distribution (i.e., the variation of the number of hydrometeors at
each diameter D), also called ''particle size distribution'', such that N ð D Þ dD is the
number per unit volume of hydrometeors having a diameter from D to D þ dD.
Based on observations of raindrops, snow crystals, and hailstone diameters (hail
of course, may not be spherical; Figure 2.2 ), a gamma distribution is often
assumed, so that

N ð D Þ¼ N 0 D e D ð 2 : 48 Þ

where N 0 , are the intercept parameter, shape parameter, and slope

parameter, respectively. The limitations to assuming a gamma distribution must
be acknowledged: for example, bimodal distributions of hydrometeors are not well
represented by a gamma distribution. For example, when large water droplets
break up into smaller ones the distribution may be bimodal.
When k ¼ 3

, and

M 3 , a measure of the mass of the hydrometeors (because the
cube of the diameter is both proportional to the volume—which, when multiplied
by density, yields mass—and to the mass mixing ratio) is used as the only
prognostic variable, in much the same way as a scheme is referred to as a ''one-
moment'' scheme because only one moment, the third moment, is a prognostic
variable. If

;

is allowed to
vary. Different relations are used for different types of hydrometeors; the more
different classes of hydrometeors included,

¼ 0 and N 0 is assumed to be a constant, then only

the more like a bin scheme the

parameterization becomes.