DEPARTMENT OF COMMERCE
SINCLAIR WEEKS, Secretary
WEATHER BUREAU F. W. REICHELDEREER, Chief
MONTHLY WEATHER REVIEW
JAMES E. CASKEY, JR., Editor
Volume 83 Number 3 MARCH 1955 Closed May 15, 1955 Issued June 15, 1955
ON THE NUMERICAL PREDICTION OF PRECIPITATION
J. SMAGORINSKY
Numerical Weather Prediction Unit, U. S. Weather Bureau, Washington. D. C.
and
G. 0. COLLINS I
Short Range Forecast Development Section, U. S. Weather Bureau, Washington, D. C.
[Manuscript received March 14. 19551
ABSTRACT
With the three-dimensional field of velocity predicted by numerical methods it is possible to predict the moisture
distribution and hence the occurrence of large-scale saturation. A three-parameter model was used to predict the
12-hour precipitation for the early stages of the storms of November 24, 1950 and November 5, 1953, neglecting cloud
storage, supersaturation, a possible lack of condensation nuclei, evaporation from falling droplets, and moisture
sources. Large-scale orographic influences were taken into account.
A quantitative comparison of the predicted rainfall with the correspondingly large-scale smoothed observed
precipitation indicates a skill comparable to that of the predicted flow. An examination of the small-scale observed
rainfall indicates that in these cases convective instabilfty resulted in large standard deviations from the large-scale
average. Numerical prediction of regions of convective instability, which is also shown, could for the time being
be utilized for subjective interpretation.
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3.
4.
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7.
8.
9.
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No. 4, Bultland, Md.
341693-6"1
1 Present affiliation: Joint Numerics1 Weather Prediction Unit, Federal Offloe Buildlng
1. INTRODUCTION
Workers in the field of numerical prediction have con-
cerned themselves almost exclusively with the prediction
of changes in the three-dimensional mass field and hence,
as a direct consequence, changes in the large-scale velocity
and temperature fields. The prediction of these elements
is necessary though not sufficient for the prediction of the
large-scale precipitation fields. Thus onefinds in the
literature (e. g. [7, lo]), for the most part only qualitative
comparisons between numerically predicted vertical mo-
tion fields and observed precipitation.
The prediction of precipitation is a difEcult task mainly
for two reasons: (i) a lack of detailed knowledge of the
physics of formation of cloud particles and their precipita-
tion; (ii) the fact that, unlike the normal situation with
the other meteorological elements such as pressure,
63
54 M O N T H L Y W E A T H E MABCH 1955
temperature, and wind, small-scale precipitation often is
of much greater magnitude than the large-scale precipita-
tion.
Our object will be to devise a dynamical large-scale
precipitation model which by the application of numerical
methods will enable us to predict. In what follows we will
circumvent (i) by assuming [14] (a) there are always suffi-
cient condensation nuclei, (b) no supersaturation, (c)no
supercooling, (d) no non-adiabatic processesasidefrom
those resulting from changes of state, and (e) and (f)
cloud storage and evaporation from falling droplets are
both negligible compared with significant amounts of
precipitation. Furthermore, wewill assume no moisture
source outside of the atmosphere, i. e., no evaporation
from the surface of the earth. The justification, for
shorter periods, lies in the fact that evaporation as a
function of space is generally of much smaller amplitude
and much more uniform than condensation, although in
the large the two must balance? It is obvious that the
present laminar lower boundary condition used in numer-
ical prediction is incapable of permitting the transport of
moisture across the lower boundary. In order to take
evaporation sources into account we must ultimately
assume a turbulent boundary layer to make possible
eddy diffusion normal to the boundary.
DifEculty(ii)will, for quantitative purposes, be ig-
nored, but will be discussed again later. It must be
pointed out that as a consequence of the large non-linear
interaction, even the large-scale precipitation calculations
must be in error.
2. CONDENSATION
If T is the mixing ratio, then during the condensation
process drldt<O, so that the condensation rate per unit
volume is -pddrjdt, where pd is the density of the dry
air. Thus the rate of precipitation reaching the ground
is
0, r<rs
$= { dr, dt -2 T=T,
dt
Equation (1) may be transformed to
P=- JDwdr, D: {
1
S P W r=rs
where w=dp/dt and D is the domain of integration. A
schematic typical plot of w/gp, vs r is shown in figure 1.
Only the shaded area thus contributes to the integral.
Hence the precipitation may be computed if r , w, T and
p , are known.
In order to predict precipitation it is thus necessary to
predict the moisture field as well as the three-dimensional
mass field. This may be accomplished by the require-
ment of continuity of T :
b r -d r br bt -d t V-Vr-w - bP
""
where V is the horizontal wind vector and the velocity
components and partial derivatives are taken with respect
to a coordinate system in which x is positive eastward,
y is positive northward, and p is positive downward.
Thus with V , w, and r known as functions of x, y, and p ,
the transport terms may be computed. To calculate
drJdt, which under our assumptions is the condensation
rate, we must look to thermodynamic principles.
The mixing ratio may be defined in terms of the vapor
density (or absolute humidity) p*gby
P*=rPd (6)
Assuming the water vapor to behave as a perfect gas we
have that
e mR
R*T m* p*=--, p="-
where e is the vapor pressure over water or ice depending
on whether T is 20" C. or <O" C., respectively, m* is
the molecular weight of water vapor, m the molecular
p,+, being the density of liquid water (1 gm. ~m .-~), po
the pressure at the earth's surface, and T , the saturation
mixing ratio, a function only of p , the pressure and T,
the temperature. In the absence of evaporation processes,
the integral is different from zero only where drJdt<O.
The precipitation accumulated over a period r is thus
given by
F=J"r Pat (3)
W. F. Libby has reported [9] that raindrops from hurrimnea and stratiform clouds am
on the average 3 weeks old. FIGUEE: 1.-A schematic plot of--w/(gp,) vs. r.
MARCH 1955 MONTHLY WEATHER REVIEW 55
weight of dry air, R is the gas constant for dry air, and
hence R*=1.608 R. It is assumed that the temperature
of the vapor is the same as that of the dry air. Under
saturated conditions r=r,, and hence
d In rs-d In (e,/T) d In pd
dt dt dt
" "
(8)
where e, is the saturation vapor pressure.
The Clausius-Clapeyron equation to good approxima-
tion may be written as
...
d In e,"rd In T=O, 7s- AR*T
L
in which L is the latent heat of condensation and A" the
mechanical equivalent of heat. For T<OC. L should
be replaced by the latent heat of sublimation.
Thus (8) becomes
d In rII d In T d In pd -(r-1) --- (10)
"
dt dt dt
During the condensation process the equivalent potential
temperature, e,, is a conservative property.
d In eE-
dt
"
0
where
In eE=const--K In pd+(l-K) In T+a,
7
J
and c, is the specific heat of dry air at constant pressure.
Equation (1 1 ) yields
d In r, d In pd
d In T- dt dt (13)
a!"K-
"
dt a !-(l -K )
if the individual change of L is assumed small. Thus the
individual change of temperature may be eliminated
between (10) and (13) to give
Since in these considerations the air is saturated, A is
purely a function of p and T. Hence, as one expects, the
condensation rate is proportional to the compressibility.
For p=700 mb. and T=OC.: a=.0493 and r=19.7.
One is then justified in assuming a<<1 and y>>2, so
that
Both for small- and large-scale motions dpd/dt is given
to good approximation by
thus
d In r8
dt bP
"
-A- b In W) w<o
Since A and bln pd/bp>o, the side condition in (16) is
required to insure that d In r8/dt<0 for the condensation
process.
Equations (l ), (2), (5)) and (16) taken together with a
knowledge of V and w sate for the prediction of precipi-
tation, given as initial conditions the spatial distribution
of mass and mixing ratio.
With the approximation that w.=wbp& where w is
the vertical velocity when the height, z, is a vertical co-
ordinate, that -= -g p d by the hydrostatic approxima-
tion, and that the lapse rate is moist adiabatic, (16)
reduces to a result deduced by Fulks [8]. Fulks did not
intend to use this result prognostically but rather to cal-
culate precipitation by means of an analogue of equation
(1 ) from a known field of r, T, w, and p . However, we
have shown equation (4) sufficient to calculate contem-
porary precipitation.
aP bz
3. VERTICAL MOTION
The large-scalefield of vertical motion is not an ob-
servable quantity. However, application of the hydro-
static and geostrophic filtering approximations to the
primitive hydrodynamic equations permits one to deduce
the vertical motion field given only the three-dimensional
mass (or pressure) field and appropriate boundary
conditions. This can be seen from the following differ-
ential equations [7] :
thermodynamicenergyequation (IS) topredictthethl.ee-dimenslonal velocity. By * There is apparent inconsistency in the usf~ of the adiabatic approximation in the
this, we assume that the heat of condensation does not materlally altar the fleld of flow
overperiods of the order 24-36 hours. However,there Is evidence that for longer
niflcant interactions [Ill. In the present situation we may regard the approxhation es
periods, all non-adiabatic mums and sinks, orography and dch friction result in sig-
a quasi-linearization with respect to the condensation in analogy with mall perturbation
theory where one assumes that the basic flow determinea the propagation of a disturbance,
but that this disturbance does not aifect the mean flow to the first order of amall quam
tities.
56 MONTHLY WEATHER REVIEW
in which we take
7" V"+f -f
1
In 8=const+" In p+ln
C
CP
j is the Coriolis parameter and c, is the specific heat of
air at constant volume.
In deriving the vorticity equation (17) the vertical
advection of Vorticity, oavlbp, and the turning of the -. . "
vortex tubes in vertical planes, - --- - have been aw a v aw bu
ax ap a y a p
neglected.
Since initially z is given as a function of x, y, and p ,
system (17), (18) and (19) may be regarded as two
dserential equations in two unknowns: &/at and a.
Customarily w is eliminated and the system solved for
bz/bt. Then w can be determined explicitly by substi-
tuting back either into (17) or (18). However, we may
alternatively eliminate bz/bt and write the differential
equation governing w :
in which we abbreviated
For simplicity, we have replaced q by.f when r ] occurs
as a coefficient and also have assumed i3 $/a p , the static
stability, to be constant in an isobaric surface, an approxi-
mation effectively made in all two-parameter models.
This three-dimensional Poisson equation may be solved
by relaxation methods, once given the three-dimensional
mass distribution which is sufficient to determine the
inhomogeneous terms, and also given boundary conditions
at p=po, p=O, and at the lateral boundaries. Consistent
with a two-parameter model w may be expressed as a
quadratic function of p . Taking
w=wo, p=po
w=o, p=o }
Equation (20) may then be written as
(24)
We have thus reduced the problem to one of the solu-
tions of a two-dimensional Helmholtz equation into which
the boundary conditions at the top and bottom of the
atmosphere already have been incorporated.
The surface vertical velocity, wo, is induced by orogra-
phy or by skin frictional action and is given by [5]
wo =Vo-Vh+b [K sin (24 To (2 5 )
where h is the elevation of the large-scale orography, To is
the geostrophic relative vorticity in the friction layer, K
is the average eddy diffusivity and Y the angle between
the wind and isobars.
4. SPECIALIZATION TO A THREE-PARAMETER
MODEL
For the purposes of actual calculation, the general pre-
diction equations are reduced to a three-layered model aa
described in general by Charney and Phillips [7] and in
particular by Charney [4]. This model is equivalent to
an atmosphere consisting of three divergent barotropic
layers.
The vertical velocity may be computed from the
thermodynamic energy equation (18) which may be
rewritten
where D/Dt is the horizontal individual time derivative
and a bar denotes the standard value. A quasi-lineari-
zation has been performed so that when the stability
appeared as a coefficient, the standard atmosphere sta-
bility is used. This is consistent with the approximations
made in deriving the prediction equations for this model.
The numerical integration scheme used here carries the
history of the motion in the potential vorticity. Hence
b +/b t is never calculated explicitly. It is thus necessary
to approximate b 4/a t in (26) by finite differences. In
numerically integrating the potential vorticity equation,
it is necessary to perform the initial time integration non-
centrally over one finite difference time interval and thence
to proceed by means of centered time differencesover
double time intervals [7]. The result is that a small oscil-
lation in 4 with a period of two time intervals is artificially
induced.* This oscillation is barely detectable; however,
then o may be expressed in terms of the vertical motion,
a*, at the level p*=p0/2 :
of the flrst barotropicpredictions [SI. The hourly forecast data wereavailable for in-
4 This was observed by one of the writers who participated in the ENIAC calculatlon
spection since the internal memory limitations of the ENIAC required the output of
all intermediate results.
MUCH 1955 MONTHLY REVIEW 57
where time differences of the b, field are taken over an
odd number of t,ime intervals, a significant error is intro-
duced. In the case of differences over one time interval,
the oscillation in some instances completely masks the
physically real finite difference approximation to the con-
tinuous time derivative. Therefore in the present calcu-
lations, all time differences are ta.ken over a double
interval.
In finite difference form (i and j are horizontal coordi-
nates, k pressure, and T time and the corresponding inter-
vals between integral values of these coordinates are As,
As, Ap, and At, respectively) equation (26) becomes
where
J
A s=300 km., At=% hr., and mi? is taken to be unity
for the latitude span used here on a Lambert conformal
projection(see for example [12]), k=1, 13, 2, 2%, 3 refer
to 200, 350, 500, 675, 850 mb., respectively, and the
boundaries k=%, k=3% are placed at 25 mb. and 1000
ab., respectively.
tu can first be computed cent,rally over a double time
interval a t t=% hour using information at t=O, and t =l
hour. The initial vertical velocities s t t =O are therefore
taken to be the same as those at time t=% hour. Inspec-
tionof the subsequent data showed that changes within
a half hour interval are sufficiently small that no serious
error is introduced by this assumption.
Since the vertical velocities computed by the model,
especially those a t 350 mb., are at rather high levels in
the troposphere and above the maximum concentration
of moisture, it is considered necessary, for the prediction
of precipitation, to deduce vertical velocities a t lower
levels. Predictions with models giving greater definition
at lower levels, e. g., a three-layer model with p2 the vertical
coordinate [4] which gives values at about 575 and 825
mh. or perhaps a five-level model, may eliminate this need
to interpolate. Given w at four levels: 675, 350 mb. and
the boundaries 1000 and 25 mb., one can fit a cubic with
respect to pressure, giving a continuous function:
'w(P) =M(p)wm+N(p)~aro (30)
in which M and N are the interpolation variables, and the
anterior superscript denotes first approximation.
As previously pointed out, the computations of vertical
velocities in this particular model are based on the assump-
tion t,hat the vertical velocity is zero at 1000 mb., the lower
boundary. This assumption may not produce serious
errors in the prediction of the geopotential field but does
become more detrimental in the prediction of precipitation.
Vertical velocities at the surface produced by forced ascent
over orographic barriers can contribute considerably to
the precipitation [lo]. The most logical and consistent
way to include the effects of large-scale varying terrain
would be to incorporate this lower boundary condition
implicitly in the prediction equations. This can be done
without great difficulty (see for example eq. (24)). Since
the flow prediction equations used here do not take terrain
into account, these effects are included a posteriori.
Vertical velocit,ies at the lower boundary, 1000 mb., are
computed from (see eq. (25))
wo=Vo*vh=Vgoo*vh (31)
where VQO, is the geostrophic wind at 900 mb. Vgo, was
extrapolated quadratically from information at 200, 500,
and 850 mb. For a second approximation to the vertical
velocity we define
"(p)='w(p, P wo 1000 (32)
While this procedure is somewhat arbitrary, it has the
characteristic, somewhat similar to that of the atmosphere,
that the effects of the lower boundary on the flow are
damped out approximately linearly with decreasing pres-
sure. Vertical velocities may be calculated in this manner
for p=400(100)900mb. denoted by k'=l to 6,
respectively.
In the integration of the finite differenceform of the
potential vorticity equation the choice of At was bounded
by the requirement of computational stability with
respect to As. Since there is no such restriction in using
the results of the integration of the potential vorticity
equation together with the finite difference form of the
system (l), (2), (5), and (16), a time interval, A f t , in
precipitation equations, may be chosen commensurate
only with the time scale of the dependent variables.
The b, field at the levels k', may be quadratically inter-
polated and extrapolated from the values at k=1, 2, 3.
The temperature @=-- - R dP ") is predicted only at
k=1% and 2%. While a linear interpolation for inter-
mediate levels may adequately determine the tempera-
58 MONTHLY REVIEW MAEOH 1955
ture, experiments in extrapolating to flanking levels
prove inadequate for our purposes. This can be over-
come to some extent by making selective use of the ob-
served detail in the initial soundings. The observed
initial temperatures at each level IC' are smoothed sub-
jectively yielding T i J k ' . The predicted temperature
change, AT, at k=1%, 2% over the time interval A't is then
interpolated and extrapolated linearly to give the tem-
perature change at the levels k'. This process may be
iterated:
T;,!$'=[T''+(AT)"+'] t j k '
(AT)~3:=[Uk~(Ar),,+bk~(AT),.]~'w } (33)
where and b k t are determined from the interpolation
formula. Since & is determined independently by a
much simpler scheme, there will be some hydrostatic
inconsistency with the corresponding Tkl field. This may
be avoided by determining Tkt as described, but integrat-
ing vertically to obtain &, using as a reference one of
the &"S.
The system of prediction equations (l ), (2 ), (3 ), (5 ) and
(16) in finite difference form is thus:
(34)
(35)
1 and rfl-l=rO for /=O
for r'>O
For computationa,l convenience the density lapse rate
is taken to correspond to that of the moist-adiabatic
lapse rate, and hence is a function of height only. This
approximation is not necessary with numerical prediction
models of three or more parameters. I n t h e present
three-parameter model the static stability, although
constant in the vertical, may vary in time and in the hori-
zontal. Hence the continuous form
in finite differences becomes
At the upper and lower boundaries &$lap must be de-
termined non-centrally. In all calculations, for con-
venience, L is taken as the latent heat of condensation,
and e and r are taken with respect to water, irrespective
of T. This restriction may easily be removed.
Comparisons of predicted geopotential tendencies with
observed 12-hour changes [2] indicate that a time interval,
A't, of 12 hours represents an upper limit. However, ap-
preciable error can be introduced by the assumption that
the saturation state is constant over so long an interval.
It is estimated that A't=3 hours would result in more
tolerable truncation errors. It is possible, however, to use
a larger Art if the truncation error resulting from the con-
stant saturation state approximation is to some extent
eliminated. This may be accomplished by estimating the
time, t,, of occurrence of change of saturation state during
A f t . One can determine from the initial mixing ratio, ro,
the saturation state at that time. In the instance where
To<(, then one may determine whether saturation will
occur during A't from the non-central form of (37) between
t =O and t,:
t , is thus defined and determinable. If t,>Art then drldt=O
over the entire interval. If not, dr,Jdt computed from
(38) is to be weighted by the factor (1-tJA't). For
initially saturated conditions oneuses a correspondingly
derived criterion for the occurrence of unsaturated condi-
tions during Art. Thompson and Collins [14] used a simi-
lar technique for precipitation calculations in which A't
was taken as 12 hours. They did not take horizontal
moisture t.ransport into account.
5. THE EXPERIMENT
The sit,uations chosenwere the early synoptically
unspectacular stages of two cases of rapid cyclogenesis.
The initial conditions were taken at 0300 OMT, November
24, 1950, and 1500 GMT, November 5 , 1 9 5 3 . The object
was to predict the accumulated precipitation over a 12-
hour period. For this purpose A't was first taken as 12
hours. With the scheme outlined in the previous section,
the only predictive element is a determination of the time
of occurrence of change of saturation state. Otherwise,
only the initial w and r fields are needed. A secondcal-
culation was prepared in which A't was taken as 3 hours
and, therefore, the full set of prediction equations were used
in 4 iterations for a 12-hour forecast.
Numerical flow forecasts for the same two periods
UARCH 1955 MONTHLY WEATHER REVIEW 59
treated here were performed by the Princeton group using
a three-parameter 900-700400-mb. model [4], but
3-hourly vertical motion fieldswere not calculated. The
fact that these situations were highly baroclinic in the
lowest levels meant that the 850-500-200-mb. model was
incapable of recognizing the large amount of potential
energy available for conversion to kinetic energy. The
result is that the predicted development by the 850-500-
200-mb. model was inferior to that by the 900-700-400-
mb.model in both cases. This is particularly true for
the November 5, 1953, case.
6. DISCUSSION OF RESULTS
For the purposes of verifying the forecasts of large-
scale precipitation accumulating over 12-hour periods,
observations from all hourly precipitation reporting sta-
tions were utilized. This of course gives a relatively fine-
grained picture of the actual precipitation-with stations
having an average separation of roughly 30miles. The
observations were taken from Climatological Data for the
United States [15] and from specially prepared listings
supplied by the National Weather Records Center in
Asheville, N. C.
NOVEMBER 24, 1950
A comparison of the 850-mb. maps at initial time 0300
GMT (fig.2A) and 12 hours later (fig. 2E) shows that the
depression centered south of Lake Superior filled but a
secondary trough was forming to the south. This is
predicted by the modelused(fig. 2E). At 500 mb. the
Low centered in Wisconsin (fig. 2B) elongated and moved
to southern Lake Michigan (fig. 2F). Again this evolu-
tionwas adequately predicted (fig. 2F). At 200 mb. the
flow forecast was not quite as good, but has less bearing
on the present problem and is not shown. The calcu-
lated vertical motion fields at t,he beginning and end of
this 12-hour period at 675 mb. and at 350 mb. are shown
in figure 2 (C, Dl G, and H). The predicted fields of 6
and w at the intermediate 3-hourly intervals are not
shown.
In figure 3A are plotted all non-zeroandnon-missing
12-hourly precipitation reports to the south and east of
the heavy boundary. The results using At=12 hours
(dashedlines) indicate a maximum of .23 inch at Lake
Erie. However for At=3 hours (solid lines) there occurs
a double maximum, one at Lake Erie of the same magni-
tude and another in Tennessee with somewhat larger
meximum value. A comparison of non-zero points and
predicted precipitation greater than .01 inch indicates a
good qualitative forecast for either model and thus a
fairly reliable statement of the time of onset and cessation
of precipitation. However examination of the individual
reports indicates large deviations even from the results
usingAt=3 hours. Figure 3B summarizes the large-
scale average observed precipitation and the standard
deviation from this average. This was constructed by
averaging all observations within the isohyetal channels
of the prediction in whichAt=3 hours. These points
were computed from over 1,300 observation stations in
eastern United States. The model is seen to have over-
predicted the large-scale precipitation by a factor of
approximately 2 for the intermediate amounts and by a
factor of 1.25 for the maximum.
It is also important to bear in mind that the three-
parameter modelused here (850-500-200-mb.)gives a
flow forecast inferior to that of the 900-700400-mb.
model even in the first 12 hours.
I t is evident from a comparison of figure 2 (C, Dl G,
und H) with figure 3A that upward vertical motion is
only a necessary though not a sufficient condition for
precipitation even for qualitative purposes. For in-
stance one would not have judged from the vertical
velocity field alone that the primary precipitation maxi-
mum would be in Tennessee.
NOVEMBER 5, 1953
This case deals with a nascent cyclone in the northeast
Gulf of Mexico at 850 mb. (fig. 4A) which deepened and
traveled up off the east coast to 31 N. latitude in a
12-hour period (fig. 4E). Actually the 850-500-200-mb.
model gave a forecast which was quite poor in describing
the observed development. No deepening and only
slight eastward motion were predicted at 850 mb. (fig. 4E);
At 500 mb. the primary trough in northeastern United
States (fig. 4B and F) was predicted to move too rapidly
to the east and the development of a closed circulation
wasmissed entirely. On the other hand the secondary
in the Gulf, associated with the 850-mb. closed Low, was
predicted to move to the east at the approximate speed
observed.
I t should be noted that the anticyclogenesis predicted in
north-central United States at 850 mb. and 500 mb. but
not observed does not directly affect the area of our
precipitation forecasts.
In figure 5Awe have, as before, the raw precipitation
observations superimposed on forecasts computed using
At=12 and 3 hours.
An essential difference in the observed precipitation
between this case and that of November 1950is the oc-
currence of large amounts along the Carolina and southern
Virginia coasts. The observed 850-mb. flow at the begin-
ning and end of this period showsonshore winds. This
coastal effect is well known [I] but is not taken into account
by the present theory. Fresumably this could be accom-
plished by an extension of the theory resulting in equation
(25) to apply to variable surface roughness.
A significant difference is observed between the pre-
dictions using At=12 hours and 3 hours. For At=12
hours the maximum is only half as large as that for At=3
60 MONTHLY M a B C H 1955
FIGURE 2.-November 24, 1950 case: initial, predicted, and verifying horizontal flow (A, B, E, F,); initial and predicted vertical flow
(C, D, G, H) for the 12-hour period 0300-1500 GAIT, November 24, 1950.
MONTHLY WEATHER REVIEW 61
Figure 2"Continued
341883-55-2
62 MONTHLY WEATHERREVIEW MAECH 1955
FIQURE 3.-(A) Observed precipitation during period 0300-1500 GMT November 24, 1950 with computed isohyets (At=3 hr., solid and
At=12 hr., dashed) superimposed. R denotes precipitation 20.01 in. but amount is unknown and thus R is not included in the
numerical verification. Missing reports and zero precipitation are not shown. (B) Summary of results with observations smoothed
for large-scale verification. Numbers in parentheses at plotted points give number of stations averaged in the respective isohyetal
channel. Dashed lines indicate standard deviation of observations in the respective channel.
MARCH 1955 MONTHLY WEATIIER REVIEW 63
FIGURE 4.-November 5, 1953 case: initial, predicted, and verifying horizontal flow (A, B, E, F); initial and predicted vertical flow (c
D, G, H) for the 12-hour period 1500 mrr November 5 to 0300 ~M T November 6, 1953. See p. 64 for parts E, F, G, H.
64 MONTHLY REVIEW
MAEOH 1955
MARCH 1955 MONTHLY W E A T H E R R E V I E W 65
-- . hours and is also displaced to the north. Had the flow
. ' prediction for this period been correct, as essentially
resulted from the 900-700400-mb. model, the maximum
. ' would have been elongated northeastward along the coast.
, However one can speculate that, in spite of this elongation,
the maximum would not have been reduced because of the
larger vertical velocities associated with the increased
icular case the vertical motion fields (fig. 4C,
Dl G, and H) would have given a fairly realistic qualita-
tive indication of the region of precipitation without
referring to the moisture field and its changes. On the
other hand, the magnitude of the vertical velocity would
have been misleading. In the November 24, 1950 case a t
675 mb. an average maximum vertical velocity of"6.5
cm sec" yielded a predicted maximum precipitation of
.23 in./12 hr., while in the November 5, 1953case an
average maximum vertical velocity of"2.5cmsec"
gave .43 in./l2 hr.
' In figure 5B we have a comparison of the large-scale
smoothed observed precipitation vs. the computed. In
this case the plotted points, representing a total of 184
observations within the Verification area chosen have
more-or-less uniform scatter about the perfect forecast
is interesting to observe the occurrence of much larger
standard deviations than were found in the first case.
This is a normal characteristic of precipitation at lower
latitudes where the incidence of convective instability
is greater.
184 STATIONS
, o .m .x) .Y) .a .s a BO .m ea 90
OBYRVED PRECIPITATION 1IN.p 12HR)
B (Mean within predicted Jsahyets with class interval .IO inches per 12 hr.)
WGURE 5.-(A) Observed precipitation duringperiod1500 GMT
November 5 to 0300 QMT November 6, 1953 with computed
isohyets (A'6=3 hr., solid and A't=12 hr., dashed) superimposed.
R denotes precipitation 20.01 in. but amount unknown and thus
R is not included in the numerical verification. Missing reports
7. PREDICTION OF RELATED ELEMENTS
Charts of the predicted and observed 12-hour changes
in dew point at 900 mb. were prepared for the November
5, 1953 case (fig. 6). Cursory qualitative examination of
the field distributions reveals an excellent correspondence.
The 7.5' C. rise in western Kentucky is predicted quite
well. Although the fall area in the Great Lakes region is
correct, the minimum is predicted to be too far east.
Howemr, although tho secondary minimum in western
Virginia is correctly placed, the predicted minimum is too
small in magnitude. A large discrepancy is noted over
Arkansas where the observed rise is predicted as a sub-
stantial fall. (.ln the other hand the negative band through
the Gulf States, Oklahoma, Kansas is correct, although the
observed minimum in the Gulf States is predicted farther
to the northwest and twice too large. The rise off the
southeast Atlantic Coast is correct but one-third the
andzero precipitation are not shown. (B) Summary of results observed magnitude. Fortunately the discrepancies
with observations smoothed for Iarge-scale verification. Num- pointed out above do not greatly affect the precipitation
bers in parentheses at plotted points give number of stations
averaged in the respective isohyetal channel. Dashed lines calculations*
indicate standard deviation of observations in respective channel. The details of the moisture Prediction Caldations show
66 MONTHLY W E A T H E R R E V I E W MARCH 1955
FIGURE 6.-Predicted and observed 12-hourchange of dew point
(C.) at 900 mb. between 1500 GMT, November 5 and 0300 GMT,
November 6, 1953.
that on the average, the condensation, the local time
derivative of mixing ratio, and the horizontal and vertical
transports of mixing ratio are all of the same magnitude.
Hence any approximation neglecting one of these would
be invalid.
We see that the large-scale average precipitation is on
the whole predicted correctly. Thus onecould expect
to be able quantitatively to predict total precipitation
over large watersheds the scale of a unit mesharea-
approximately 35,000 sq mi. The fine structure due to
small-scale instability is not capable of being predicted
by the models used. For a wholly dynamical prediction,
small-scale non-linear theory of the type utilized by
Tepper [13] wouldbe required. The obvious difficulty
is that an extremely h e network of surface and aero-
logical observations wouldbe required in order to ade-
quately specify initial conditions. For the time being
this is not economicallyfeasible operationally, even if
the theory were adequately developed. For most pur-
poses it might besufficient if statistical moments of
precipitation higher than the mean (such as the standard
deviation) weresomehow attainable by dynamical
methods.
Thus, what suggests itself, is a statistical-mechanical
approach such as is used in other branches of physical
science. The fine structure itself is not predicted but
interpolated from
Observations
FIGURE 7.-Predicted and observed areas of convective instability
(bOs/bp>O), 0300 GMT, November 6, 1953.
rather the distribution of the statistical properties of the
fine structure. Even for this modest requirement, it is
necessary to understand the mechanism of convective
processes in a moist atmosphere and the dependence of
the small-scale dynamics on the ambient large-scale
conditions.
From parcel stability considerations we have the well
known result that a necessary condition for convective
instability is that the equivalent potential temperature
decrease with Figure 7 shows a 12-hour forecast
from the 1500 GMT, November 5 , 1953 situation of the
occurrence of convective instability according to the
parcel criterion. I t is seen that the verification with the
observed occurrence is quite good,especiallysince the
flow prediction at this time was already beginning to
degenerate. The release of this instability could be
accomplished, theoretically, by sufficient lifting.
8. FRICTIONAL VERTICAL VELOCITY
Although the influence of frictional divergence is not
included in the prediction model, it is of interest to study
the magnitude of the vertical velocity so produced in one
of the cases treated. Figure 8 shows the frictional vertical
recent is an article by 0. H. B. Priestley, Buoyant Motions and the Open Parcel,
8 Parcel dynamics have often been viewed critically, and with good reason. The most
7he Metcorolopicul Magazine, vol. 83, No. 982, April 1954, pp. 107-114.
MABCH 1955 MONTHLY WEATHER REVIEW 67
FIGURE 8.-Frictional vertical velocity (cm. sec. -*) at top of friction
layer computed a t 1500 GMT, November 5, 1953.
velocity at the top of the friction layer computed from the
second term of equation (25) using the observed 850-mb.
00w at 1500 GMT November 5, 1953. The constants
arechosen according to average values suggested by
Brunt [3]:
K=10 m.2 sec. and ~~2 2 .5  .
We see that the magnitude of the forced frictional
vertical velocity can attain values of over 1 cm.sec..
Assuming that this boundary influence decreases linearly
with decreasing pressure (as was done for the orographic
influence), the computed free vertical velocities even at
675 mb. would change by not more than 25 percent.
However there may result significant shifts in the location
of centers of maximum or minimum. For example the
frictional maximum of + 1.5 cm.sec. south of Talla-
hassee would result in shifting the free maximum at 675
mb. to the west over the northern Florida peninsula with
little change in magnitude.
On the other hand, fields of free vertical velocity not
associated with large-scale storms, and which are thus
moreof the magnitude of frictional vertical velocities,
could undergo large percental changes, conceivably chang-
ing sign. Under proper moisture conditions this could
make the difference as to whether or not small amounts
of precipitation would result over large areas.
9. CONCLUDING REMARKS
The foregoing represents a rudimentary physical frame-
work by which the occurrence of precipitation may be
predicted. The reality of the results may be thought
of as depending in part on the goodness of the numerical
three-dimensionalflow prediction and in part on the
physical model accounting for moisture changes and the
prccipitation process.
We have seen that the precipitation calculations were
not too sensitive to the failings of the flow prediction in
the two cases studied. To some extent this must be due
to the time-wise integrating process (eq. (3:)). Since these
were only 12-hour predictions, it is difficult to surmise the
influence of a continually deteriorating flowprediction
over a 24-hour period, On the other hand the predictions
of dew point and convective instability, whichdo not
involve such an integration, apparently have not suffered
inordinately.
The approximations regarding the condensation process
discussed in the Introduction do not appear to be crucial
in these cases. A more comprehensive study for a larger
number of cases should disclose any systematic effect on
the predictions. Relaxation of the constraints introduced
by these approximations depends on progress in the field
of cloudphysics. Without these constraints it should in
principlebepossible to distinguish betweenlarge-scale
cloudiness and clear skies.
The approximations whichwere made only for com-
putational convenience in that portion of the calculations
done by hand, may of course be eliminated in program-
ming the entire problem for a high-speed computing
machine.
Judging the results of the two caseschosen by the
subjective standards normally used for precipitation
verification, we conclude a skill comparable to that of
the predicted flow.
REFERENCES
1. T. Bergeron, The Problem of Artificial Control of
Rainfall on the Globe, 11. The Coastal Orographic
Maxima of Precipitation in Autumn and Winter,
Tellus, vol. 1, No. 3, August 1949, pp. 15-32.
2. B. Bolin and J. G. Charney, Numerical Tendency
Computations from the Barotropic Vorticity Equa-
tion, Tellus, vol. 3, No. 4, November 1951, pp.
3. D. Brunt, Physical and Dynamical Meteorology, Cam-
bridge University Press, Cambridge, 1939.
4. J. G. Charney, Numerical Prediction of Cyclogene-
sis, Proceedings of the National Academy of Sciences,
vol.40, No. 2, February 1954, pp. 99-110.
5. J. G. Charney and A. Eliassen, A Numerical Method
for Predicting the Perturbations of the Middle
Latitude Westerlies, Tellus, vol. 1, No. 2, May
248-257.
1949, pp. 38-54.
68 MONTHLY mTEATHER REVIEW MABCH 1955
6. J. G. Charney, R. Fjgrtoft, and J. von Neumann,
Numerical Integration of the Barotropic Vorticity
Equation, Tellus, vol. 2, No. 4, November 1950,
7. J. G. Charney and N. A. Phillips, Numerical Inte-
gration of the Quasi-Geostrophic Equations, for
Barotropic and Simple Baroclinic Flows, Journal
of Meteorology, vol. 10, No. 2, April 1953, pp. 71-99.
pp. 237-254.
Scale Heat Sources and Sinks on the Quasi-Station-
ary Mean Motions of the Atmosphere, Quarterly
Journal of the Royal Meteorological Society, vol. 79,
12. R. J. List (Editor), Srnithsonian Meteorological Tables,
6th Revised Edition, Smithsonian Institution,
Washington, D. C., p. 485.
13. M . Tepper, The Application of the Hydraulic Anal-
NO. 341, July 1953, pp. 342-366.
8. J. R. Fulks, Rate of Precipitation from Adiabatically
Ascending Air, Monthly Weather Review, vol. 63,
No. 10, October 1935, pp. 291-294.
9. W. F. Libby, Address to American Chemical Society,
September 12, 1954.
10. J. S. Sawyer, A Study of the Rainfall of Two Synoptic
14. J
-.
Situations, Quarteily Journal of the Royal Meteoro-
logical Society, vO1* 787 No* 336) 1951 PP- 15. U. S. Weather Bur&&, mimatological Dab f o r the 23 1-246. United States by Sections, vol. XXXVII, No. 11,
11. J. Smagorinsky, The Dynamical Influence of Large- November 1950.
ogy to Certain Atmospheric Flow Problems,
Weather Bureau Research Paper, No. 35, October
1952, 50 pp.
C. Thompson and G. 0. Collins, A Generalized
Study of Precipitation Forecasting. Part 1: Com-
putation of Precipitation from the Fields of Mois-
ture and Wind, Monthly Weather Review, vol. 81,
NO. 4. April 1953. PP. 91-100.