DEPARTMENT OF COMMERCE
CHARLES SAWYER, Secretary
WEATHER BUREAU
F. W. REICHELDERFW, Chief
.MONTHLY WEATHER REVIEW
Editor, JAMES E. CASKEY, JR.
Volume 79
Number 7 JULY 1951
~~~~~~~~~ ~ ~
Closed September 15,1951 Issued October 15,1951
L
ON
.
FORECASTING CEILING LOWERING DURING CONTINUOUS
LOUIS GOLDMAN
U. S. Weather Bureau Airport Station, Boston, Mass.
[Original manuscript received November 2, 1949; revised manuscript received April 13 19511
RAIN
ABSTRACT
During steady rain, the ceiling lowers in a discontinuous fashion. The ceiling heights may be predicted with
sufficient accuracy by using a set of empirically determined rules. To obtain a relation for the time of occurrence
of these ceilings, the factors which influence cloud formation are considered. An expression is derived for the rate of
moisture increase due to evaporation from falling raindrops. The rate of moisture change, given by this expression,
is combined with the effect of the other factors in order to obtain a formula which may be applied to find the time a
ceiling of given height will occur. The variables in the forecast formula are (1) the wet-bulb temperature depression
measured before the start of rain and (2) F,, the effective rate of moisture increase caused by factors other than
evaporation. Values for F, are found empirically. An approximate method, based on the surface value of the
depression, is used for finding the time of occurrence of the 800-, 500-, and 300-foot ceilings. This approximate
method appears to be best suited for forecasting the 500-foot ceiling.
formula for the rate of downward growth of the fracto-
cumuli forming underneath the rain cloud during steady
precipitation. His derivation is based on the assumption
that all rain forms by the melting of snow falling out of
the nimbostratus, accompanied by cooling of the air just
beneath the cloud. This coolingcauses a steepening of
the temperature gradient below the zero isothermal, con-
sequently increasing convection wbich results in the for-
mation of the fracto-cumuli. Assuming that the heat
required to melt all the precipitation comesfrom the
surrounding air, Findeisen shows that the rate of down-
ward growth of the low cloud is given by:
2 =6 .2 N d t
where N is the rate of rainfall in mm.hr." and defat is
the rate of lowering in cm.sec.".
According to equation (1) clouds formed in rain lower
at a rate directly proportional to the intensity of rainfall.
An attempt to apply the result to forecast ceilings during
1 The definition of the term "ceiling" is subject to change. In order that no confusion
shall result, it will he employed here to mean the height, above the ground, of the bsse of
the lowest cloud layer covering more than half the sky. This deflnition approximates
no specification as to the amount of cloudiness is intended, terms such as "base height of
the official meaning in effect at the time of the ceiling data used in this report. When
cloud (or cloud layer)" or "height of base of cloud" will be employed.
133
134 M O N T H L Y W E A T H E JULY 1951
TIMEAFTERSTARTOFRAIN(HR)
0
c
c: U.6
d cn X c -4 Y
W 0
0
4 8 TIME AFTER START OF REN(HR.)
TIMEAFTERSTART OF RAIN(HR)
FIQURE 1.-Variation of ceiling and rain intensity in three typical cases observed at
Boston. (A) April 24,1944. (B) May 15,1944. (C ) March 7, 1945.
rain in eastern United States leads to failure, perhaps
because conditions influencingcloud formation are
somewhat different from those assumed by Findeisen.
Three typical cases of ceilinglowering during rain,
observed at Boston, are shown in figure1. The ceiling
does not lower continuously, but appears to remain fixed
until a cloud forms below this height and increases in
amount sufficiently for its base height to become the
ceiling. This ceiling then remains practically constant
until another cloud layer, closer to the ground, appears
and increases in amount so as to constitute the ceiling.
In this manner the ceiling lowers discontinuously until a
finalcloud layer appears close to the ground. This final
cloud layer may increase in amount and extend downward
NEAREST CLOUD HEIGHT INDICATED BY RADIOSONDE CTHSD. FT.) . .-
FIGURE Z.-Observed ceiling during rain plotted against the nearest level defined by
rules (1) and (2) found from latest radiosonde preceding the rain. Portland, Maine,
1945-1947.
very slowly until the damp air is displaced by drierair.
The smoothed curves through the points representing
ceiling and the corresponding time it was first observed
are typical of those obtained for all cases and may be
taken to define the variation of ceiling during continuous
rain. Comparing the ceiling and rain intensity, it is seen
that the ceilinglowers at a decreasing rate as the rain
intensity increases, and that the minimum ceilingis
reached before maximum rain intensity. Apparently
the rate of ceiling lowering and rain intensity are nega-
tively correlated.
With these observational facts in mind, it is the purpose
of this study to examine rules which may be usefulin
forecasting variations of ceiling during rain, and, through a
logical approach, to develop an objective method for
predicting the rate of ceiling lowering during rain.
BASE HEIGHT OF CLOUD LAYERS DEVELOPING
IN RAIN
Two generally wellknown rules are applied bythe
aviation forecaster to determine the base height ofcloud
layers likely to occur during rain. These are :
(1) The base of a cloud layer will be at a height &There
the temperature lapse rate changes from positive to less
positive or negative.
(2 ) The base of a cloud layer will be at a height where
the wet-bulb temperature depression, or dewpoint
temperature depression, is a minimum.
To test these rules the Portland, Maine, surface reports
for the years 1945,1946, and 1947 were examined for all
cases of continuous rain. All the differing ceilings reported
during each rain period and all the heights defined by rules
(I) and (2 ) found from the latest radiosonde preceding the
JULY 1951 MONTHLY WEATHER REVIEW 135
rainwere listed. In a fewcaseswhere multiple heights
were indicated within a short distance, the lowest height
was selected. Due to an incomplete file of radiosonde data
available, several cases were omitted, leaving a total of 32
cases for study. All the ceilings reported in these cases are
shown plotted against the nearest height given by the
radiosonde in figure 2 . If the two rules gave both the
necessary and sufficient conditions for determining the
height of cloud bases, all the points in figure 2 would fall
near the straight line OA. However, all the points do not
fall near OA, and if the rules are still considered sufficient,
then the deviations must be due to (1) errors in measuring
or estimating the ceiling, (2 ) time lag from the radiosonde
observation to the time of ceiling observation, or (3) con-
tinuouslowering of ceiling. The points above OA, of
course, can not be in error because of continuous lowering.
The greatest deviations occur in the points above OB,
determinedfrom radiosonde data preceding the rain by
more than 9 hours; it is likely that a stratum of higher
temperature or humidity appeared at these levels in the
meantime. The smaller deviations included between OA
and OB relate to cases with little or no lag in observation
time and can be attributed to the unavoidable error of
estimating or measuring the ceiling.Since the error of
estimate is as likely to be positive as to be negative, all the
points included between lines OB and OC may becon-
sidered to agree with the rules for forecasting the height of
cloud bases and, in particular, the ceiling during steady
rain.
A relatively large number of points fall near or below
1,000 feet. Since there is a large separation between these
points and the clouds above them, it is unlikely that the
points correspond to ceilings which lowered continuously.
Furthermore, the error in the measurement of the base
height of low clouds is generally negligible. These ceilings
are reported regardless of the time lag between radiosonde
observation and start of rain, so it must be assumed that
a temperature inversion develops near the surface after the
start of rain, if not sooner. The wind almost always in-
creases with the approach of continuous rain. Mechanical
turbulence associated with the increasing wind results in a
temperature inversion not far from the ground. A maxi-
mum relative humidity at the base of the mechanically
producedinversion and the inversion itself may not be
evident in a radiosonde taken before the start of rain.
Strong winds result in higher inversions and a more rapid
development of the inversion so that the number of ceilings
between 1,000 and 2,000 feet not associated with an inver-
sion is smaller than the number below 1,000 feet.
If the few cases lying between line OD and the 2,000-
foot level are attributed to a lag in observations, then the
ceilings falling between OC and OD may be due to down-
ward growth of a cloud base. However, most of these
points are close to, or above, OC, so considering the
possible error in observing ceilings, the continuous lowering
of ceiling during rain is negligible in most cases.
This leads to the following additional rules which may
be found useful in forecasting the variation of ceiling
during rain:
(3) If the rain is of sufficient duration, a ceilingwill
occur below 2,000 feet. Most frequently it is a ceiling
of 800 feet.
(4) During continuous rain a ceiling generally does not
occur at the height of temperature discontinuity and/or
maximum humidity until after the occurrence of a ceiling
corresponding to the next higher level of temperature
discontinuity and/or maximum humidity.
(5) The ceiling remains practically constant until the
next lower cloud layer appears and increases sufficiently
for its base height to become the ceiling.
Applying the above rules to available radiosonde data
leads to a reasonably accurate forecast of the ceilings
which will occur during continuous rain. From the study
of ceiling variation it is obvious that if the radiosonde ob-
servation is taken over 6 hours before the start of rain
then a significant ceiling may occur which is not given by
the rules. However, whether or not such a ceiling can
occur may be determined readily by inspection of radio-
sonde dat,a closer to the rain area, or if these are not
available, by noting the base heights of clouds reported
in the rain area.
Since the ceilings which occur, when the rain is of suffi-
cient duration, can be found with the degree of accuracy
required in an aviation forecast, it remains to develop a
method of forecasting the time these ceilings will first
occur.
FACTORS INFLUENCING TIME O F CLOUD
FORMATION
From the manner in which the ceiling varies during
continuous rain, it appears that the important factors
influencing the variation may be:
(1) Advection of warmer and more humid a i r a t selected
levels. In these strata of warm humid air which appear
in advance of the rain area, the relative humidity increases
upstream, reaching the 100 percent value a t the forward
edge of the cloud sheet. The cloud itself may not move
at the speed of the wind because other factors associated
with the rain tend to increase the relative humidity of
the air in the strata.
(2) Vertical mixing. Mechanical turbulence at the
boundary between the warm stratum and the colder air
beneath it and in the layer next to the ground causes
vertical mixing which tends to increase the moisture con-
tent of the upper part of the mixed layer at the expense of
the lower part. If the moisture content of the mixed
layer is sufficiently high, a cloud will form near the top
of the mixed layer, as shown by Petterssen [2 ]. The base
height of this cloud will remain constant until the mois-
ture content, due to other factors (e. g., evaporation and
aclvdction), increases sufficiently to lower the mixing con-
densation level (MCL) . Thus, if successively lower layers
136 MONTHLY WEATHER REVIEW JULY 1951
form in intervals of a few hours, the gradual lowering of
the cloud basis, except the lowest, is negligible when the
ceiling variation is considered.
(3) Evaporation from falling raindrops. Evaporation
is effectivein increasing the relative humidity of the
entire air column. The rate of evaporation is greater
when the dryness of the air is greater. Evaporation
moistens the dry air between the moist strata rapidly, but
evaporation diminishes as the relative humidity increases,
and unless the rain is warmer than the wet-bulb tempera-
ture of the surrounding air, evaporation alone cannot
produce condensation. When the air is very dry in the
lower layer, evaporation determines the time of formation
of the lowestclouds. Vertical mixing may produce an
inversion near the ground, but if the air is dry, the hfCL
will lie above the layer of mixing and no cloud will form;
however, evaporation will increase the amount of moisture
rapidly-the drier the air, the more rapid the increase.
Eventually the MCL falls within the layer of mixing and
a cloud forms near the base of the turbulence inversion.
This cloud builds downward as evaporation continues.
The problem of determining the time it would take for a
cloud to form due to the combinedeffect of advection,
vertical mixing, evaporation and, perhaps, other factors
is a complex one. However the problem may be simplified
by confining attention to evaporation and allowing for the
other factors by inclusion of a suitable parameter. Since
the effects of vertical mixing and advection are highly
correlated, each depending on the wind and moisture dis-
tribution surrounding the rain area, a single parameter
may suffice for the effects of the two.
MOISTURE INCREASE DUE TO EVAPORATION
If a falling raindrop is conceived to be surrounded by a
thin viscous air film, through which heat is transferred by
conduction, and this boundary layer to be surrounded by
a turbulent zone, through which heat is transferred con-
vectively, then a relation for the transfer of heat between
the raindrop and the surrounding free air may be derived
readily. Thus, Newton’s Law, which is applicable to the
boundary layer, may be written:
” “-hfa(Tb- T?) at (2 )
where dQ/dt is the rate of heat transfer through the film,
a is the mean area of the film (which may be taken to be
the surface area of the drop) , T , is the surface temperature
of the raindrop and T b is the temperature at the outer
boundary of the air film. The heat conductance h, is
defined by the ratio kl6, le being the conductivity of the
film(which may be taken to be that for air) and 6 the
thickness of the film.
Taking equation (2 ) to define the “iilml’ conductance of
heat h,, analogous equations are written to define the
“convective” and “over-all” conductances, h, and hg,
respectively, thus:
s=h,a(T-TT,) dt
and
--hh,a(T-TT,) d Q at -
where T is the temperature of the free air.
If there is a continuous flow of heat between the rain-
drop and the free air, then i /hB =I/h,+i/h,. By consider-
ing the dimensions of the variables upon which h, and 6
depend, i. e., the raindrop diameter d, its speed relative to
the air v, the air viscosity p , and the air density p, it can be
shown that:
where R=- is, by definition, the Reynolds Number for
the raindrop. The form of the function may be found
experimentally; however, except for a shape factor, which
will be assumed to have the value of unity, the function
may be approximated by the empirical relation:
dv P
Ir
@=0.45+0.33(R)0.5e k
which is based on the correlation of data for the flow of air
at right angles to the axes of single cylinders ranging in
diameter from 0.001 to 0.375 inch [3]. Values of hg for
the range of sizes found in rain, computed by equation
(5) and based on R values calculated by Gunn and Kinzer
[4] and k=0.0000568 cal.cm.”sec.”(OC)-’ are listed in
table 1. For sizes ranging from 0.05 to 0.50 cm. there is
only a 12 percent variation of individual values from the
mean value of 0.0042 c. g. s. units so it may be assumed
that hg is constant for all raindrops.
TABLE 1.-Terminal velocity, Reynolds number and over-all heat
conductance for raindrops of various size
d *
Cm. 0.05 .10
.15
.25 .XI
.35
.30
.40 .45 .50
Cm. sec. -1
206 403
65. 7
541 269
649 542
742 12.39
866
806 852 1613 1991 883
900
2357 2704 909 3033
!r calculated from the m
CaZ. see.-’ cm.-2 deg.-1
0.0046 ,0046 .0045
,0043
. w1
I
s (GUM and Kinzer 141 ). ‘Equivalent drop diamete
?Terminal velocity of fall for distilled water droplets in stagnant air at a pressure of
XReynolds number = (air density) X (equivalent diameter) X (measured velocity) +
..
760mm., temperature 20’ C . andrelative humidityof 50percent (Qunnsnd Kinzer(41).
(viscosity of air). (Gunn and Kinzer [4].)
Now, if it is supposed that the heat required to evaporate
rain comes from the surrounding air and that a state of
equilibrium is reached instantaneously, then the tempera-
JULY 1951 M O N T H L Y W E A T H 137
ture change dT, of a unit mass of air in the time interval
dt, is
dT=--- hgA (T-Tw)dt (6)
C P
where T, is the wet-bulb tempe.rature, c, the specific heat
for air at constant pressure, and A the total raindrop area
in unit mass of air. Neglecting variations in c,, it remains
to find the variation in A, in order to solve equation (6).
Lenard [5] measured the drop distributions in various
types of rain and expressed the measurements in terms
of the number of drops of each size falling on a unit
horizontal area in a unit time. Using these data, the
value of A near the ground may be computed from:
A=- ?r Z(T) nidi2 P
where ni is the number of drops of size d i and terminal
velocity Z I ~ falling on a unit horizontal area in unit time.
Values of A computed from this equation, together with
a description of the rain and its intensity, as given by
Lenard, are shown in t8able 2.
TABLE 2.-Character of rain and surface area of raindrops
I I A l
I n t e n s i t y of rain
Mm. min.-1
(1) 0.09
(2) .06 (3) .ll
(4) .05
(5) .32
(6) .72
(7) .57
(8) .34
(9) .26
Veryordinary looking rain. Cm. 2 gm.-l
0.0083
.m7
.0058 Breaks occurred during which the sun shone.
DO.
,0045 Beginning of a thundershower. ,0115 Sudden rain from a small cloud. ,0292 Violent rain like a cloudburst, some hail. .0220 Heaviest pehod, less heavy period, and period ,0230
.0085 took the form of a cloudburst. of stopping of a continuous fall which at times
There is a good correlation between the rain intensity
and A, however the intensity values given by Lenard
appear to be computed from the size distribution, which
may account for this good correlation. To find the true
relation between rain intensity and A, it is necessary to
have data for measured rain intensity. These data are
not available, so the type of rain will be considered instead.
At the beginning of rain, when the air is relatively dry,
as in cases (3) and (4), A has a value from 0.004 to 0.006
cm2gm". The values increase to about 0.008 or 0.010
after the rain becomes steady, as in cases (1) and (a),
and then remains at that value until stopping, case (9);
but during rain of cloudburst intensity A may reach as
high as 0.03 cm2gm". Since the minimum ceiling is
generally observed to occur before the rain has reached its
. maximum intensity, it seems reasonable to assume A has
the constant average value 0.005.
Equation (6) may now be solved readily to give:
10.0
8.0
6.0
c
U 4.0 L z
cn u)
W
0
a f= 2.0
0
W
a 3 I-
U a 1.0 W a 5 0.8
0.6
c
W
-I 3 m
0.4
5
I I I 1
I
0.2 ' I I I I I I , I 0 2 4 6 8 IO
TIME AFTER START OF ,RAIN (HR.)
FIGURE 3.-Wet-bulb depression plotted against hours of rain for three cases of continuous
rain at Boston, Mass.
and T ~= (T-T,), is the value of the wet-bulb depression
measured at the start and T=(T-T,) is the value a t
the end of the time interval t.
Using the values found for A and hE, and c,=0.24
cal.gm." gives:
W = (0.0042) (0.005) (3600)/(0.24)=0.3 h,-'
If factors other than evaporation may be neglected,
then w may be computed directly from equation (8).
These factors may be assumed negligible when the air is
dry at the start of rain and the wind is light during the
rain. However surface variations of depressiononly are
available and, since the diurnal variation at the surface
may be appreciable, the diurnal factors would have to be
considered. The normal diurnal variation of the wet-
bulb depression shows an almost constant value during
the night, so to determine w from the surface variation
it is best to select cases in which rain began during late
evening.
A case in which evaporation appears to be the factor
controlling the wet-bulb temperature depression occurred
at Boston on May 6, 1942. Rain began a t 1930 EST.
The wind was SSW 15 to 20 m. p. h. until 2 hours before
the start of rain when it diminished to less than 10 m. p. h.
and then remained gentle for the remainder of the night.
The wet-bulb depression plotted against hours of rain
on semilogarithmic paper (fig. 3) gives a straight line of
slope 0.30 h -l , thus verifying equation (8) and the value
for w found indirectly. Two other cases plotted in the
138 MONTHLY WEATHER REVIEW JULY 1951
FIGURE 4.-Nomograph based on equation (10) giving t. as a function of R and 70. The
average values of F, given in the table were determined from Portland, Maine, data
using equation (12) or (13) (see fig. 5).
same manner can also be fitted with straight lines of the
sameslope. In these cases the windwas moderate to
fresh, but steady, and, no doubt, advection and vertical
mixing were appreciable.
TIME OF OCCURRENCE OF A CEILING OF GIVEN
HEIGHT
If rain is at its equilibrium temperature then, according
to equation (8), evaporation alone can not result in the
formation of a cloud, since t= 00 when 7=0. However
evaporation results in an exponential decrease in the de-
pression, so it may well determine the time of cloud devel-
opment, at least at times when the air is relatively dry
at the start of rain.
Suppose factors, other than evaporation, cause an
independent decrease in the wet-bulb temperature depres-
sion, say F , per unit time, then the total rate of change
of the depression during rain is
If it is assumed that F, varies with height only, then
the time t,, after the start of rain, when the air at a given
height becomes saturated is
t,=- log, (1 +E T o )
1
W
where 7 0 is the value of the depression at that height a t
the time rain begins. Equation (10) is represented
graphically in figure 4.
If the depression is measured t' hours before rain starts
and has a value r r at that time, then since F , has been
assumed constant, the depression at the start of rain is
TO=T'-Fztr (1 1)
which may be subsitiuted in equation (10) to find t,.
However, if d<FJ', then the time of saturation, t'8, in
hours after the time r' is measured is simply:
FIGURE 5.-Values of 3'. by elevation determined from Portland, Maine, data using
equation (12) or (13).
Obviously use of equations (11) and (12) willunder-
estimate the time of saturation; the error will be greatest
when 7' is measured longbefore the start of rain, i. e.,
be,fore F, becomes constant. In most cases F, can not be
determined directly, so it will be considered sufficient for
this study to obtain some empirical values suitable for
forecasting the time a ceiling of given height willoccur.
Such values may be computed readily from a combination
of equations (10) and (II), i.e.,
or from equation (12), when the time of saturation is close
to the time rain begins. Now, if the time a ceiling of
given height first occurs is taken for t , (or t'J then the
values for F , determined by using equations (12) or(13)
may be considered appropriate corrected values for finding
the time a ceiling of given height will occur.
Such values of F, were found from the Portland, Maine
data and are shown plotted in figure 5. Evidently the
points fall in two classes; in one, F, is relatively large, and
in the other it is small. From the winds aloft last reported
near the start of rain, it was found that the smaller set is
associated with winds having a direction of 210° to 270°,
and the larger set of values associated with a wind direction
of 20' to 200°-except for a few cases.
It appears that the higher values correspond to cases in
which the air trajectory passes either over water or close to
the center of the rain area; while the lower values corre-
spond to cases in which the air has a land trajectory not
passing close to the rain area center. For places on the
North Atlantic Seaboard, such as Portland and Boston,
when continuous rain is associated with secondary Lows
traveling northward, the higher F, values should generally
be applicable. The smoothed values given in figure 4 a,re
JULY 1951 M O N T H L Y W E A T H E R 139
TIME(HR1 AFTER 2200 EST RADlOSOND€ OBSERVATION
I I I I I I I I I -10 -5 0 +5 t10
T-TEMPER)ITUREVCI
FIGURE 6.-(A) Temperature (T) and wet-bulb temperature depression (7') from
with observed ceiling at Portland, Maine, April lC-11,1946.
Portland, Maine, sounding, 2200 EST, April 10, 1946. (B) Comparison of computed
based on insufficient data to be considered very reliable,
however they may be used until better estimates are made.
To illustrate the application of the results obtained so
far, we now consider an example forecast of ceiling dur-
ing continuous rain based on upper air data.
EXAMPLE OF USE O F UPPER AIR DATA TO FORECAST
CEILING DURING RAIN
Rain began a t Portland 0015 EST, April 10, 1946.
The lapse rates of temperature and wet-bulb temperature,
as determined from the 2200 EST radiosonde of the 9th,
are shown in figure 6 . At 2300 EST, the pibal reached
9,000 feet, indicating winds of 150' to 210' at all levels.
With these data, the forecast of ceiling is made as follows:
The temperature and depression lapse rates indicate
that ceilings of (a) 9,000 feet, (b) 6,700 feet, and (c) 4,50@
feet will occur if the rain is of sufficient duration. Ceilings
of less than 4,500 feet are not indicated, therefore the
other heights selected are (d) 800 feet, (e) 500 feet, and
(f) 300 feet. The 800-footceiling height is selected
because it has been found to be the most frequent low
ceiling. The 500- and 300-footceilings are chosen for
their significance to aviation.
Considering each ceiling in turn:
9,000 feet,
From the depression lapse rate, ~'=0.9' F.
From the table in figure 4, F2=2.1' F. h r -I .
Since F z t ' =(2 .1 ) (2 .2 5 )>~' , equation (12) is applied:
t',=r'/F2=O.4 hrs. after the observation of T', i. e.,
the 9000-foot ceiling will occur at %'2%'4 EST.
-
6,700 feet.
From the depression lapse rate, ~' ~3 .6 ' F.
From the table in figure 4 , F,=1.7' F. hr-l.
F2t'=(1.7) (2.25)=3.8>~', so t',=3.6/1.7=2.1 hrs.
4,500 f e e t .
~' =3 .2 , F 2 =1 .4 ; F ,t ' =3 .2 =~' ; therefore the 4,500-
foot ceiling occurs at the time rain starts, i. e., a t
0016 EST.
after 2200 EST, or at 0006 EST.
800 feet.
~' =3 .5 , F z =0 .4 ; FZt'=0.9<T'; therefore applying
equation (1 1 ): ~~=~' "F ~t ' =2 .6 .
t I i 0 1 I I I I I I I 1 .I 1 1 -I 1
00 02 04 06 .OS IO 12 ' 14 16 'IS 20 22 00
TIME OF DAY (E.S.T.)
FIGURE 7.-DiurnP variation of wet-bulb temperature depression.
From figure 4 , for ~~=2 .6 and F,=0.4, it is found that
t,=3.5 hrs., i. e., an 800-footceilingwilloccur
3.5 hrs. after the start of rain, or at 0546 EST.
~' =2 .8 , F ,=0 .1 6 , 7 0 =2 .4 , t ,=5 .7 hrs. after rain
(e) 500 feet.
starts, i. e., a t 0600 EST.
(f) 500 feet.
r f =2 .4 , F ,=0 .0 8 , T0=2.2, and t,=7.4 hrs. after
A comparison of this computed variation of ceiling
rain starts, or at 07.45 EST.
with the actual ceiling variation is shown in figure 6 .
ESTIMATE O F WET-BULB DEPRESSION IN LOWER
LAYERS FROM SURFACE VALUES
Moisture gradients in the lowerlevels are usually
variable in advance of rain, and sinceradiosonde data
are relatively scarce, it is desirable to make use of surface
data to predict the time of occurrence of lowceilings.
I t has been mentioned that the diurnal variation of the
surface depression may be appreciable and shouldbe
considered. Surface values before the start of rain may
not be a good measure of the value a few hundred feet
from the ground, where the diurnal factors may beless
important.
Generally, the range of the diurnal variation increases
with the dryness of the air, so there should be very little
diurnal variation during rain. However, near or before
the start of rain, diurnal factors may cause a large varia-
tion. Thus, comparing the depression variation before
the start of rain with the normal daily variation in figure 7 ,
it is seenwhen rain begins in the early afternoon, the
depression increases its value by about 75 percent from
early morning to near noon; when rain begins near
midnight, the depression lowers during the afternoon and
evening a t a rate not very different from the normal
lowering.
A rough measure of the depression above the layer of
diurnal influence may be made by assuming that the
actual variation is proportional to the normal and applying
a correction to the surface depression depending upon
the time of day. These correction factors would, however,
be unreliable when applied to depressions measured
at night because there is very little correlation between
the minimum surface depression reached at night and
140 MONTHLY WEATHER REVIEW JULY 1951
the upper air values. In order to relate surface depres-
sions with upper air values, it is necessary to determine
the intensity of the factors contributing to the diurnal
variation. However, these factors, no matter how in-
tense, have little or no influence on the surface depression
at the time of day when the depression is normally at
its average daily value. At these times of day, the
surface depressions are best correlated with upper air
values.
The times of day when the wet-bulb temperature
depression has its average values may be determined
readily by averaging 24 hourly surface depressions and
finding the time of day this average occurs on the smoothed
diurnal variation. Approximate average values obtained
in this manner, but by using the differencebetween
average hourly temperature and wet-bulb temperature
values [6] for Washington, D. C., are shown in table 3.
TABLE 3.-Time! of day wet-bulb depression has i f s d a i l y average
value, Washington, D. C. (Eastern Standard Time)
I I
Month A.M. P. M. Month A.M. P. M.
Jan 1030 2130
July 0930 24330 Mar loo0 2230 Aug 0930 2030 Apr 0830 2130
June 0930 2030 Feb 1100 2230
May 0930 2030
Month A.M. P. M.
Sop o900 1930
NOV 1030 2030 Oct 0930 2ooo
Dec 1030 2030
Annual lo00 2030
___- -
Thus, the depression is normally at its mean value. at
1000 EST and again at 2030 EST. Since upper air
observations are made at about 1030 and again at 2230
EST, it follows that the morning radiosondes are best
suited for finding the correlation of upper air depressions
and surface values. This is made obvious in figure 8,
which contains scatter diagrams relating the wet-bulb
depressions at the surface, 500 feet, and 1,000 feet for
both the morning and evening observations taken within
12 hours before the start of rain. Further, since the
1030 EST points in both diagrams fall reasonably close
to a straight line of unity slope, the vertical gradient of
the wet-bulb temperature depression below 1,000 feet
before rain starts, is nearly zero at the times of day when
diurnally varying factors do not contribute to the wet,-
bulb depression.
LL t
c
A
wET-eae OEPRESSION AT SURFACE CFI
FIQUBE 8.--Scatter diagrams for cas88 0-12 hours before start of rain, relating wet-bulb
temperature depression at (A) 500 feet and surface, and (B) 1,000 feet and surface.
(Portland, Maine)
Thus, if rain starts soon after 1030, or 2030 EST, the
surface value of the wet-bulb depression at these times
may be assumed to give, approximately, the value of
the depression up to 1,000 feet. This value may then be
substituted for r’ in equation (11)tofind the depression
at any levelbelow 1,000 feet at the time rain begins, if
the appropriate value for F, is used. I t should be noted,
however, that if rain begins long after 1030 EST (or 2030),
the average values for F, given in the table in figure 4
may be appreciably higher than the actual average, so
that their use may lead to an under-estimate of ro above
the ground, in such cases.
FORECASTING TIME O F CEILING OCCURRENCE
USING SURFACE WET-BULB DEPRESSION
To illustrate the application of surface elements meas-
ured before the start of rain to predict the time low ceil-
ingswill occur, the ceiling heights of (a) 800 feet, (b)
500 feet, and (c) 300 feet will be chosen for consideration.
If t,he higher values for F, are chosen from the table in
figure 4, then equations (1 1) and (12), for each level
become:
(a) 800Jeet. If r’>0.4tf, then
r0=r’-0.40t‘ (IW
where r‘ is the surface wet-bulb depression measured at
1030 EST, or 2030 EST, whichever is closest to the time
rain starts, and t’ is the number of hours to the start of
rain. The number of hours, after the start of rain,
when the 800-footceilingwill first occur can then be
found from figure 4 using the value of ro given by equation
(Ila) and the value F,=0.40.
If 7’ 10.4t’, then the time the givenlowceiling will
first occur, in hours after the measurement of r‘, is
tS’=2.5r’ (124
Similarly for the other levels:
(b) 500feet. If r’>0.16t’, then
and if 7’ <O.l6t’, then
(3) S00feet. If r’>0.08t’, then
and if r’<O.O8t’, then
r ,,=r f -0 .1 6 t ’ (1 l b )
t’,=6.3rf (12b)
r~=r’-O.OSt‘ (1lc)
t‘,= 12.57’ (12c)
These formulas, together with figure 4, were tested on
all cases of continuous rain which occurred at Boston
during the years 1944 and 1945, assuming that the values
of F, found for Portland apply. Cases in which low
ceilings occurred before the start of rain were omitted.
The average of the wet-bulb depressions measured at
0930, 1030, and 1130 EST (or 1930, 2030, and 2130 EST)
was taken for r’, in order to reduce possible errors in the
measurement of the depression. For cases in which rain
beganclose to 1030 or 2030 EST, the last two measure-
ments before the start of rain were averaged so that the
later data could be used. In any case, the average time
of the averaged measurements to the time of beginning of
rain was used for t’. The resulting forecasts are compared
JULY 1951 MONTHLY W E A T H E R R E V I E W 141
I I I 1 1 I
0 4 8 12 16 20
COMPUTED TIME OF CEILING OCCURRENCES (HR)
FIQWRE 9,"Comperison of forecast with actual time of ceiling in rain at Boston, Mass.,
1944 and 1945.
with the time that the ceilingwas first reported, at or
below the given height, in figure 9. In most cases the
error is within 3 hours.
Points A, B, C, and D in figure 9 represent the only
800-footcases for which t' is more than 8 hours. For
all other cwes t' is 8 hours or less. The large deviations
in cases A, B, C, and D may be attributed to the error
resulting from the application of the average F, value
over too great a time interval. If the surface values
before the start of rain are used to give a better measure
for the wet-bulb depression at 800 feet at the start of
rain, the points A, B, C, D translate to A', B', C', D',
respectively.
The two points E and F show the largest departure
in the opposite direction. These correspond to a single
case. Actually the ceilingloweredfrom 900 feet to 200
feet within 2 hours just ahead of a warm front, preceded
by light winds and drizzle. I t is to be expected that when
a front lies in the vicinity of the station, before the time
rain can produce lowcloudiness, the properties of the
front, with respect to lowclouds and fog, determine the
ceiling, and therefore should beconsidered in an actual
forecast.
In many cases the ceiling did not lower to the selected
levels, so not all the forecasts made appear in figure 9.
To show how the formulas apply to all cases, the contin-
gency table 4 was prepared. This table compares the
lowestceiling observed in each case with the minimum
ceiling that would have been forecast if the time of ending
of ceiling lowering was known; for example, the time dry
air advection begins or the time rain ends. Since the
969933--51-2
computed minima are within a few hundred feet of the
observed lowest ceilings, in most cases, it appears that
conditions favoring saturation did not continue a sufEcient
time for the cloud to develop at the selected level.
TABLE 4.-Computed and observed minimum ceiling in rain at
Boston, Mass., 1944 and 1945
l l
Number of cases forecast
1 I I
/Less 400 feet than1 500 400 feet to 1 800 600 feet to 1 0yrtm I 1
2
32 3 7 11 11 Total z
2 2 0 0 0 Over 800 feet. 11 1 5 4 1 6w to800feet ...........
g 5
9 0 1 5 3 400 t o 500 feet ........... 5%
10 0 1 2 7 Less than 400 feet. - -. -.
..........
g5 .............
""-
CONCLUSION
I t is evident from this study that it is possible to predict
the rate of ceilinglowering during rain, objectively and
with a reasonable degree of accuracy. Although several
important factors influencing cloud formation have either
been neglected or roughly approximated in the derivation
of the forecast method, a logical approach was attempted
in order that the forecaster mcty at least make an approxi-
mate allowance for the effect of these factors, when known
or extrapolated. The following list, although incomplete,
suggests how forecasts made from figure 4 may beim-
proved qualitatively by the forecsster and, of course,
suggests the lines along which further study shouldbe
made in order to improve the accuracy of ceiling forecasts :
(1) If a front or trough, with which lowceilings are
associated, lies nearby, then the ceiling may lowermore
rapidly than forecast.
(2) In cases of heavy rain or snow, near the beginning
of precipit,ation, the ceiling will lower more rapidly than
forecast.
(3) With strong turbulence near the ground the time of
formation of very low clouds may be underestimated,
while the time of formation of clouds near the top of the
mixed layer may be overestimated.
(4) Ceilings may not lower as rapidly as forecast in cases
of intermittent or showery precipitation.
(5 ) The time a ceiling of given height will occur may be
underestimated if there is only slight advection of moist
air at that level; if there is dry air advection, the ceiling
may rise (e. g., near the time rain ends).
Perhaps the greatest improvement in forecasting the
time of occurrence of the very low ceilings may be attained
by (a) using F, values indicated by wind and wet-bulb
depression distributions, and (b) a study of frontal char-
acteristics.
ACKNOWLEDGMENTS
The author wishes to acknowledge the encouragement
and helpful advice given during the course of this study by
Mr. C. F. Van Thullenar, Mr. Paul H. Kutschenreuter,
142 MONTHLY WEATHER REVIEW JULY 1951
and the members of the forecast staff at the Boston Weath-
er Bureau Office. Special acknowledgment is due Mr.
Conrad 9. Mook, who presented parts of this paper with
his study of ceiling lowering a t Washington as a joint
workbefore the American Meteorological Society 171.
Unfortunately, other commitments have prevented Mr.
Mook from completing his report in time to be included
with this paper.
REFERENCES
1. W. Findeisen, “Die Entstehung der 0”-Isothermie
und die Fraktocumulus-Bildung unter Nimbostratus,”
Meteorologische Zeitschrift, Band 57, Heft 2, Feb-
ruar 1940, s. 49-54. Translation by F. A. Berson,
“The Formation of the 0”-isothermal Layer and
Fractocumuli Under Nimbostratus,” Meteorological
Offices, London. Copy available in U. S. Weather
Bureau Library.
2. S. Petterssen, Weather Analysis and Forecasting, Mc-
Graw-Hill Book Co., New York, 1940.
3. Chemical Engineers Handbook, McGraw-Hill Book Co.,
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4. R. Gunn and G. D. Kinzer, “The Terminal Velocity of
Fall for Water Droplets in Stagnant Air,” Journal of
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Band 21, Heft 6, Juni 1905, s. 249-262.
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Washington, D. C.,” Technical Paper No. 8, January
1, 1949.
7. Louis Goldman and Conrad P. Mook, “Forecasting
Ceilings During Continuous Prewarm Front Rain,”
paper presented at 98th National Meeting of the
American Meteorological Society, Washington, D. C.,
April 20-22, 1948;