__ - - __
JAKUARY 1963 MONTHLY TlJEATHER BEVIEW 13
ELEMENTARY THEORY
MOTIONS AND
OF ASSOCIATIONS BETWEEN ATMOSPHERIC
DISTRIBUTIONS OF WATER CONTENT
EDWIN KESSLFR, I11
The Travelers Research Center, Inc., Hartford, Conn.
[Manuscript recejved August 2, 1962: revised October 24, 19621
ABSTRACT
Continuity equations are used to clarify relationships between air motions and distributions of accompanying
precipitation. The equations embody simple modeling of condensation and evaporation with the following assump-
tions: (1) water vapor shares the motion of the air in all respects; (2 ) condensate shares horizontal air motion, but
falls relative t o air a t a speed that is the same for all the particles comprising precipitation a t a particular time and
height; (3) the cloud phase is omitted.
After a review of one-dimensional models, the distributions of condensate in two-dimensional model wind fields
are discussed with regard to instantaneous evaporation of condensate in unsaturated air and t o no evaporation.
The most nearly natural cases must lie between these extremes. The mcthods for obtaining solutions are instructive
of basic interactions between air motion and water transport. The steady-state precipitation rate from a saturated
horizontally uniform updraft column is shown to equal the sum of the vertically integrated condensation rate and a term
that contains the horizontal divergence of wind. The latter term becomes relatively small as the ratio of precipitation
fall speeds to updrafts becomes large. A basis for some studies of precipitation mechanisms, the equation N(l74-w)
=const., where Nis the number of particles comprising precipitation a t a particular point in space and time, Vis their
fall velocity, and w is the updraft, is shown to imply violation of continuity principles unless variatioiis in w are quite
small. Continuity equations are applied to radar-observed convective cells (generators) and their precipitation trails,
and to radar-observed precipitation pendants (stalactites), and provide bases for estimating the strength, duration, and
vertical extent of the associated vertical air currents. The stalactite study also discloses how horizontal variations of
precipitation intensity arise during precipitation descent through a saturated turbulent atmosphere.
The continuity equations are powerful tools for illuminating fundamental properties of wind-water relationships.
The conclusion discusses attractive paths along which this work should be extended.
1. INTRODUCTION
The study or time-depenclent relationships between
wind fields and water distributions derives from the belief
that knowledge of m-ind-water relationships is essential
for an intelligent approach to the numerous hydro-
~neteorological problems which hold the increasing direct
interest of' mankind. Use oE wind-water relationships in
meteorological analysis should assist the interpretation of
radar and satellite observations. Knowledge of the rela-
tionships between wind and water fields should assist our
consideration of means to modify the weather, since the
distributions 01 water are interwoven with the distributions
of latent and sensible heat and the scale, intensity, and
shape ol convective processes.
This report discusses the application of continuity
equations to several interesting problems. Some previ-
ously published material (Kessler [4] and [5 ], and Kessler
1 Rfost of the work reported here was substantially completed while the author was
employed at the Weather Radar Branch, Ocophrsics Research Directorate, Air Force
Cambridge Research Laboratories The preparation of this paper has been supported
by the U S Army Electronics Research and Development Laboratories under Contract
DA 36039 SC 89099. The results given in Scctlons 6 and 7 were first presented at the
Nrnth Weather Radar Conference at Kansas City, Mo., in October 1961, and were printed
rn the Proceedings of that conference.
and Atlas [6]), is briefly reviewed to give coherence t o
this paper, but the emphasis here is on previously
unpublished work.
The principal assumptions have been: (1) Water vapor
shares the motion of the air in all respects. (2) Condensate
shares the horizontal motion of the air but falls relative to
the air a t a speed V. Vis a negative parameter t'hat niay
vary with height but that is constant with time a t any
given height. (This is a great simplification of cloud
physics processes-precipitation having a fall velocity V
is assumed to forin as the result of condensation without a
cloud phase. At, any particular height, all of the precipita-
tion particles fall at the same speed.) (3) The moisture
capacity of the air is a function of height only. (4) The
air density is considered locally steady and horizontally
uniform.
These premises lead to a continuity equation for 144, the
density of water substance in all its phases minus the
saturation rapor density, viz.,
14 MONTHLY WEATHER REVIEW JANUARY 1963
where u and v are the winds in the z and y (horizontal)
directions, w is the wind in the z (vertical) direction, and
p is the air density. G is a generation tterm that denotes
the amount of water condensed from a unit volume of
saturated air for each unit vertical distance of air travel;
G=-p(dQ/dz), where Q is the saturation mixing ratio of
water vapor in air (for derivation of equation (I), see
[4] Section 2 ).2
When 144 is negative, it shares the motion of the air
and represents the amount of moisture that must be
added to saturate the air; when M is positive, i t falls
relative to the air at a speed V, and represents the amount
of condensate in saturated air. (3 plus A4 is the total
water content. These associations between M and V
imply instantaneous evaporation of condensate in un-
saturated air, a Sact clearly perceived when one considers
that the term MV=O wherever M <O . The descent of
condensate contributes a t a rate Il.ilrV to the addition of
34 in unsaturated air just beneath, the rate OS addition
being unaffected by the magnitude of M in the drier air
untilM>O. Only when fM>O can there be a Iallout of
water substance.
By treating condensate and vapor separately in two
equations, it is practical to study the situation in which
precipitation once formed does not evaporate in sub-
saturated air. This is discussed further in section 5b
below.
I
i
2. THE DISTRIBUTION OF M ALONG VERTICALS
WHERE THERE IS NO HORIZONTAL ADVECTION
This section summarizes results presented in several
other places, and is included here to facilitate under-
standing of the new results presented in following sections.
A. V-j-w EVERYWHERE LESS THAN ZERO
(MOTION OF CONDENSATE EVERYWHERE DOWNWARD)
(1) Steady date solutions.
Omission of compressibility simplifies the discussion
without affecting the principal conclusions, and the
following expression for the distributions of 34 in time and
height is therefore considered .
When V is everywhere the same, a condition most
closely approached in snow, the third term in equation
(2 ) is zero. The steady-state vertical profile ol' M in
a saturated atmosphere is then given by
(3)
a When A4 and 0 are in mixing ratio units, equation (1) is the same except that the
last term becomes --MV@ln p /a z ), and G=--dQ/idz. Density units arc used in this
study because radnr reflectivity characteristics, visiiul appearance, and certain physical
effects arc best understood in such tcrms.
M
FIGURE 1.-One-dimensional time-dependent and steady M-distri-
butions in downdrafts (left) and updrafts (right), when G is
constant. Values on the abscissa refer t o the total water content
minus the saturation vapor content in gmJm.3 when H= 1 km.
and G =l gm. m.-3 km.? Vertical velocity w=(4 wmaZ/H)X
[z- (z 2 /H )].
This integral has been evaluated exactly for several
values of V and a parabolic vertical distribution of
updraft w, with w=O at 2=O, H. A t._vpical result
with N(H)=O is shown in the right side of figure 1,
taken from [5]. A discussion of the application of (3) to
the descent of condensate in saturated downdral'ts is
given toward the close of section 7.
When V=O, the steady-state solutions we independent
of the shape of the updraft distribution. Equation (3 )
then gives, for updrafts, the liiniting lorm OI the distri-
bution approached after a very long time with the
unrealistic condition that condensate indefinitely retains
negligible J'alling speed. The application to downdrafts
of equations (2 ) and (3) with V=O has a iairly reasonable
basis; the solution there represents the distribution of
saturation deficit attained after sinking motion has
continued for a very long time. This solution for constant
G is illustrated in the left side o l figure 1.
When V is variable, the third term in (2 ) iiiust be re-
tained and this equation's steady-state form is then not
readily solved analytically. In such cases, finite difference
approximations have been used to obtain the solutions.
Figure 2 shows computed steady-state profiles 01 M in the
model snowstorm situation indicated by table 1. These
profiles suggest that some of the upper layers observed by
radar in winter storms may be due to the kinematic princi-
ples discussed here, rather than to the widely discussed
generator and trail mechanism (Marshall [IO], for ex-
ample), which, i t is agreed, is the significant factor in most
such observations. Since increases of fall velocity are
generally associated with increases in particle diameter
(to whose sixth power radar is sensitive), a layer of en-
hanced radar reflectivity is not likely to accompany a
layer of enhanced water',content that is due to the diver-
gence of precipitrttion fall velocities. This and several
JAXGARY 1963 MONTHLY WE ATHER RE VIEW 15
Height (km.)
-~_____
........................ 0
...................... 1..
2 3.
4
5..
6..
i
8
9 IO-..
........................ .......................
........................
......................
......................
........................
........................ ........................
....................
I
Approxi- Relativc Part,icle mate updraft 7' CX104 fall pressure W (" C .) (gmJm.4) speed
(mb.) (cm./scc.)l
1015 0 0 16. 8 100 900 30 -7 12.3 i 5
795 64 -6 12.0 i 5
700 84 -4 12.6 75 F15 96 -9 9. 7 50 540 100 -17 6. 4 50
355 F4 -43 0.8 50
470 96 -26 3.9 50 410 84 -35 1.9 50
310 36 - 50 0.4 50 265 0 -53 0.3 50
i Y
c
0 I 2 3 4 5 6
M (pm/m3)
FIGURE 2.-Theoretical steady-state water-content profiles for various surface snowfall rates near the precipitation center of a winter
storm. Thc maxima occur above layers in which the particles increase their fall speed during descent. The profiles have been c d -
culated using the parameter distributions given in table 1. (From [6].)
. other steady-state variable-V cases have been treated
elsewhere (Kessler and Atlas [6]).
1 50 cm./soc. used whcre Z'< -8' C ' wherc tcmpcraturc mcrcases sbovc -8' C . lollow. ing the descent of A$ the spcrtl IS tak& ils a 11near Jncrpmo wlth temperature to 1m em ,I-
SCC. at 0' C.
(2) Timedependent solutions
In the steady cases, the vertical distribution of A I de-
picts the individual changes following the descent of &f;
Le., dLMldz=bL!/bz. When bVldz=O, d M /d z is inde-
pendent of Ad, and the steady-state profiles provide the
key to easy determination of the time-dependent solutions.
The packets of M change by an amount dM=(biV/bz)
dz while descending through d z in a time dt=dz/(V+w).
The time-dependent solutions when V is invariant can,
therefore, be constructed horn t8he steady-state vertical
profile ol M and the curve showing the height of an A4-
packet against time. Two sets of results of such calcula-
tions based on the initial condition M=O and the upper
boundary condition M (H )=O are illustrated in figure 1
and are discussed in detail in [4] and [5 ]. When the fall
velocity is a function of height, this method of solving for
the time-dependent solutions is inadequate, but it is
practical in such cases to use finite difference methods.
16 MONTHLY WEATHER REVIEW JANUARY 1963
Many properties of the variable-V solutions can be quali-
tatively assessed from general considerations supplemented
by simple computations.
B. V+w SOMEWHERE GREATER THAN ZERO
(MOTION OF CONDENSATE SOMEWHERE UPWARD)
With V constant and the absolute magnitude of V
anywhere less than the associated value of w, there is no
steady-state solution valid throughout the depth of the
updraft column. Hovever, in this case, where V is
everywhere the same, the time-dependent solutions at
points where (V+w) =O are simply M=d&,+wGt, and else-
where can be accurately- determined by a procedure only
slightly more elaborate than that described above for
determining the time-dependent solutions that precede a
steady state.
I
(See [5 ], Sec. 4.)
3. THE STEADY-STATE PRECIPITATION RATE
FROM A SATURATED, HORIZONTALLY UNIFORM
UPDRAFT COLUMN
I
The application of continuity considerations shows that
the rate of steady precipitation at the ground is not gen-
erally equal to the condensation rate in rising air vertically
ter of an area of widespread updrafts. Equality of steady
precipitabion and condensation rates is approached as
the magnitude of the ratio of precipitation fall speeds to
updrafts increases. The results presented here are an
extension of section 5 in [4].
In a steady-state horizontally uniform atmosphere, the
horizontal advection terms are zero and the appropriate
equation is:
I above, even when there isno horizontal advection at t'he cen-
I
Integration from the base 0 to the top H of the updraft
column gives
The second term on the right of (5 ) is the precipitation rate
a t the base of the updralt because at the top MFO and,
thereiore, V=O there; the third term is the condensation
rate in the vertical column of unit cross section.
The first term on the right is better understood after
integration by parts, i.e.,
The &st term on the right of (7) vanishes because w=O
at both z=O and z=H. And the equation of continuity
for air implicit in equation (1) states that h /b z in the
second term on the right of (7) is given by
Substitution of (8) and (7) into (5 ) yields
Equation (9) is a reminder that the horizontal divergence
of the wind is implicit even in the one-dimensional forms
of equation (1). The last term in equation (9) may be
positive, negative, or zero, and is largely dependent on
the ratio of characteristic fall speed to the characteristic
~p d r a f t .~ The relative contribution of the last term in
(9) is usually most important in the strong updraft
situations where M is increased in greater proportion
than w (because the fall speed of precipitation usually
increases only a little with a large increase of intensity).
From another point of view, note that it is when V is
relatively small and only slowly varying, as in snow, that
M is relatively quite large near the ground where the
divergence is negative, and that the last term in (9)
contributes substantially to the precipitation rate near
the center of areas of widespread precipitation. In
such cases the precipitation rate significantly exceeds
that defined by the condensation term alone. Any
excess of precipitation over condensation near an updraft
core is, of course, accompanied by a deficit in areas
away from the core.
In other cases, particularly where the melting zone,
associated with a five-fold increase of particle fall speeds,
coincides with the level of no horizontal divergence
of the wind, the steady precipitation rate at updraft centers
may be somewhat less than the condensation rate there.
A physical approach to the results summarized by equa-
tion (9) is illustrated in figure 5 of [4].
These results should be of some value for relating
updraft velocities to observed precipitation rates and
radar echo intensities.
4. ASSUMPTIONS IMPLICIT IN THE RELATIONSHIP
N ( V+W) =CONSTANT
The equation
N(V+w) =constant (10)
where N is the concentration of precipitation particles a t
various points along a vertical, has been used by several
3 Equation (e) is the same whether the atmosphere is assumed to be compressible or
incompressible. However, the distribution of tbe quantities therein and the magnitude
of the integrals vary somewhat with considerations of compressibility. See, for example,
Egure 1 of [41.
JAKUARY 1963 MONTHLY WEATHER REVIEW 17
investigators, including the author [3], as a conservation
law suited for a study of precipitation models. Some
published work concerning precipitation mechanisms rests
on one or more implications of this equation (lo), even
where (10) is not explicitly invoked. Although many
important limitations of (10) and its corollaries have long
been recognized, it has not been generally realized that
this equation often stands in violation of fundamental
principles of continuity.
The conditions under which (10) is valid can be
examined by reference first to
bM
derived immediately from first principles. Equation (1 1)
states merely that in the absence of horizontal advection,
the local change of M is the sum of individual and vertical
advective changes. Substitution for d-&f/dt from the one-
dimensional form of (I) gives
dM bV b ln p
-=wG-M -+Mw __. at bz az
In (12), let M=Nm, where m is the mass of each parti-
cle in a collection of N particles. Then
Suppose that combination and breakup processes among
precipitation particles are weak and that the condensation-
evaporation term contributes to the mass of individual
particles, but not to their number; i.e., precipitation par-
ticles grow or diminish in size, but are neither created nor
destroyed. This assumption may be applicable, for ex-
ample, to the further development of a packet of small
hail. Then equation (13) can be separated into two
equations :
dN bV b In p m -=--?nN --+mNw - at bz bz
and
dm
dt N -=wG.
The temporal changes of particle concentration along
the path of an individual packet axe given by
dN d2 dN dN -=-X-=(V+w) -. at at dz dz
Divide (14) by m, substitute (16) into (14), multiply by
dz, and rearrange terms to obtain
which applies along a vertical at any time if the distri-
bution state is steady, but should be restricted to the flow
following an individual packet (individual derivatives)
when the distributions are unsteady. In the idealized
case of descent at constant fall velocity through a constant
updraft, the term bV/(V+w) in (17) is zero, but there is
still a decrease in the number of particles per unit volume
because in a compressible atmosphere, effects of horizontal
divergence, measured by the third terms in equations (14)
and (17), accompany updrafts that are invariant with
height. The effect oE the third term in (17) must in nature
be more or less compensated by the tendency of precipi-
tation fall speeds to decrease as the particles descend into
air of increasing density.
On13 if w i s constant or, at least, everywhere quite small
in comparison to V, can the integral of equation (17)
without the compressibility term be represented by
equation (10). The assumption that w is constant or
everywhere small compared with Vis quite similar to the
assumption that the horizontal or vertical divergence of
the wind is small compared with the vertical divergence of
V. This may be clearly understood after consideration of
the vertical derivative of equation (10);
Equation (18) contains the term N(bw/bz), which is
properly omitted, since it is practically canceled by effects
of horizontal divergence, as shown by the equation of
continuity for air. Note that the term N(bw/bz) has
no analog in equation (4). Use of equation (IO) across a
layer where w varies implies neglect of horizontal diver-
gence and violation of continuity principles; such use
will be associated with important errors unless the mag-
nitude of air speed changes is small.
The inherent similarity of assumptions (1) that steady
precipitation and condensation rates are equal and (2)
that equations like equation (10) can be applied to the
study of precipitation, can be understood in terms of this
discussion. As fall speed increases relative to w and M,
the integral of the horizontal divergence times M in
equation (9) and the errors associated with equation (10)
become relatively small, and these assumptions are
satisfactory.
Equation (10) is valid for estimating the change of N
accompanying important changes of V which occur over
such a short interval that variations of w are negligible,
or over larger intervals in which w is everywhere much
smaller than V. The application of equation (10) is
possibly justified in the zone in which snow melts to form
rain with an approximately fivefold increase of particle
fall speed, especially when the effects of breakup of
melting particles can otherwise be included or shown to
be small. Other possible
applications are to layers in which updraft speeds are
uniform and where Nand Vcan be estimated at at least
(See sec. 7B and table 1 of [4].)
18 MONTHLY WEATHER REVIEW JANUARY 1983
one height in the layer. In no event, of course, are
equations (10) and (18) even crudely sufficient for appli-
cation over a region where w varies and (V+w) passes
through or near zero. They are not applicable, for
example, to the study of hail. Where w attains about
equation (IS), rather than equation (lo), must be em-
ployed.
It is, at least, of acaclemic interest to note that equa-
tions (14) and (15) with boundary and initial conditions
uniquely define m and N at all heights when the mtisses
of the particles are expressed in terms of their fall veloci-
ties. The relation V=-1180D”2, where V is in m./sec.
and D is in meters (Spilhaus [15]), can be used to give
V=-130(6m/a)”e (m./sec.), where 7n is in grams.
Use of this relationship with appropriate boundary and
initial conditions allows solution of (14) and (15) for m
and N where the updrarts are known at every height and
time-or, knowledge of m and N along the trajectory of
M permits the determination of updraft distributions.
Horizontal advection when present simply requires that
equations (14), (15), and (17) be applied along the non-
vertical trajectory of precipitation descending through
the atmosphere. These equations can be viewed as R
theory 01 nionodisperse-size-distributed precipitation,
applicable in a layer where breakup and combination
processes are weak and where fall velocities are every-
where larger, but not necessarily much larger, than the
updrafts.
I the same magnitude as V, a more accurate integration of
I
5. TWO-DIMENSIONAL SOLUTIONS
A. INSTANTANEOUS-EVAPORATION OF CONDENSATE IN
SUBSATURATED AIR
This section discusses time-dependent distributions of
water substance and the methods for deriving them in two
model cases where condensate fall speeds are twice the
magnitude of maximum updrafts and in two cases where
fall speeds are half the rnaximuni updrafts. Two sets of
solutions are presented for each updraft-fall speed ratio :
one set where there is instantaneous evaporation of con-
densate in subsaturated air; the other where there is no
evaporation of condensate in subsaturated air. The most
nearly natural cases must lie between these extremes.
The methods for obtaining solutions are instructive of
basic interactions between air motions and water
transport .
The following equations have been considered in con-
nection with the two-dimensional solutions :
H i
Z
FIGURE 3.-Streamlines of the wind field corresponding to equatious
(20), (2l), and (22) withf(z)=O, over the range O<%<L/2.
Equation (19) applies to an incompressible atmosphere
and equations (20) and (21) satisfy the corresponding con-
tinuity condition, viz, bu/dx+ bwldz=O. In equations
(31) and (2 2 ), f(z) is the nondivergent environmental
wind. These equations are solved in the layer 0 5 z< H ,
with &I specified at z=H. The limits of integration in
the horizontal direction are usually chosen so that there is
no horizontal wind at these boundaries, for example,
nL/2 <x 5 (n+ I)L/2, where n is an integer or zero when
f(z) =O. Then the one-dimensional solutions already dis-
cussed exist along the verticals a t the horizontal limits of
the problem area, and no boundary conditions other than
the value 01 M at z=N need be specified a t these limits.
Other dternatives are both practical and interesting but
are not discussed further here.
The wind field described by equations (20), (21), and
(2 2 ) withf(zj =O is shown by figure 3. With V constant
for M>O and V=O wherever M<0, and with initial con-
dition M=O, boundary condition M(H) =0, and G=con-
stant, equation (19) has been solved by finite difference
methods for the field of M a t various times. As noted in
the introduction, these assumptions concerning 114 and V
imply the instantaneous evaporation of condensate in any
subsaturated air into which i t falls. The top rows of
figures 4 and 5 show solutions for two relationships bc-
tween the uniform fall speeds of condensate and the maxi-
mum uDdrafts. Details concerning. the method of solu-
J-4NUhRY 1963 MONTHLY lVEA4THE1E REVIEW 19
tion and discussion of these solutions and associated
budget parameters are contained in section 5 of [5 ].
B. NO-EVAPORATION OF CONDENSATE IN SUBSATURATED AIR
Equation (19) can be solved in a modified way that
admits of no evaporation of condensate once formed.
Such a method is interesting because in subsaturated air
the most nearly natural case must lie between the ex-
tremes of no evaporation and instantaneous evaporation
of precipitation. The revised method for solving (19) is
based on the following.
Consider two sets of equations, one set for condensate
and one for vapor. Each set constitutes a restatement of
the same conservation principle expressed by (19). We
have
MZO dM -_ dzM-~+V, M>O (23b) dt --=wG, p=O (23~~);
w>O at
(244 ; a%- --Wl P<O d t
where A4 is now the d s precipitation content, always
positive or zero, and p is the vapor density minus the
saturation vapor density, zero in saturated air and a
negative quantity in unsaturated air. Note that equa-
tion (23a) is applied only in saturated updrafts (elsewhere
dM/dt=O); (23b) is applied to determine the motion of
iM. Since q=O in all saturated updrafts, equations (24a)
and (24b) are o€ concern only in following the motion and
changes of p in unsaturated air. The term MdVldz
does not appear in equation (23u) because Vis assumed
constant for condensate. In areas where M >O and
q<O, it is necessary to apply both sets of equations.
There is no evaporation when the values of M and p are
not added at points where p is negative and h!‘ positive.
The two-dimensional solutions have been determined
by combining the host of one-dimensional solutions
applicable along the trajectories (also streamlines in
these cases of steady wind fields) of M and p. Figure 6
illustrates the streamlines for V/w,,,= -2. Note that
the streamlines of p are identical with those of the wind
field, and that equations (24a) and (24b) can be combined
and intepated to yield p=po+GAz. The solutions for p
are thus given by the vertical displacenients along air-
streamlines; these displacenients have been determined
by finite-difference and graphical methods. The time-
height relationships for A4 and the steady-state profile OS
M along each M-streamline have also been determined
by combinations of numerical and graphical techniques.
The results for Mare summarized in figure 7, which is the
principal basis for the graphical development of time
dependent solutions in this case.
To obtain the no-evaporation solutions, it is necessary to
know the boundary separating air where q=O from that
where q<O; this is defined by the locus of particles that
have risen in the updraft part of the cell as much as they
had previously descended in the downdraft portion.
Figure 6 shows, for three times, this demarcation as de-
termined by finite-difference methods in the interior of
the circulation cell and by exact methods on the lower cell
boundary. Along the streamlines for condensate, growth
of M i s assumed to follow the curves of figure 7 only so
long as M is in air that is saturated. As soon as M-
parcels cross the boundary separating saturated from
unsaturated air, i.e., separating air where q=O from air
where q <O , growth of ceases and the remainder of the
M history is due to advection only. The no-evaporation
solutions where V/wrnaX= -2 are shown in the lower TOW of
figure 4.
The solutions of the problem where V/wmaX= - 112 are
shown in the lower row of figure 5 ; these have been ob-
tained in the same manner as described above, but with
somewhat greater difficulty, especially where the streani-
lines of M are loops. (See, for example, [5] figure 7.)
The solutions in upper and lower rows of figures 4 and
5 must differ only where precipitation falls into air which
has been rendered unsaturated by the air’s prior descent.
Other differences between corresponding diagrams, where
not attributable to dralting uncertainties, are due to use of
a smoothing equation in the program for digital computa-
tion of instantaneous evaporation cases. Since large
numerical differences between solutions of instantaneous-
evaporation and no-evaporation models are confined to
small areas in the cases shown, it has not seemed worth-
while to extend to the present study the detailed budget
computations discussed in [5 ].
The solutions shown can be scaled to any distribution
of atmospheric parameters which differ by constant
factors from those used here, as discussed in [5 ]. For
new scale height 94 new generating function g, new
vertical air speed ‘yKax, and new fall velocity nKaxV/
wrnaX, the new solutions denoted by di are given by
d= M g X/GH which occur at new times y= Tycazz/
wmaxH. The solutions for given G,
H, and w max can be applied to any horizontal scale, since
the equation of continuity for air requires that stretching
of the horizontal scale be associated with an equivalent
stretching of the horizontal wind speed, when the verti-
cal air speed is unchanged.
The particular wind and water fields used here have
some features demonstrably inconsistent with thermo-
dynamic principles. For example, thermal changes must
closely follow moist and dry adiabatic processes in sat-
urated updrafts and unsaturated downdrafts, respectively.
Since updrdts and downdrafts have equal intensity in our
model wind field, the descending current for a representa-
tive condit8ionally unstable initial lapse rate becomes
considerably warmer than the ascending current at cor-
responding levels, long before the circulation has pro-
gressed to the illustrated final stages. Therefore, another
model should be considered, say a radially symmetric
(See sec. 2 of [5 ].)
.
20
-)
~~~~
<!& yl('
-A
4 - -7
JAKUARY 196.3 MONTHLY WESTHER REVIEW
0 0.5 L/2 L /2
X V = -Im/sec , W max= 1/2m/sec
-
0.0
trates the case in which there is no evaporation of condensate.
The time 2700 sec. marks substantial overturning of the air in
the circulation cell. Furt,her discussion is in the text.
0.71
y 0.5
W
N 0.41
0.3 1
FIGURE 4.-Analyzed fields of water content based on a steady wind field of shape as shown in figure 3, maximum vertical air speed of 0.5
m./sec., uniform condensate fall speed of 1 m./sec., and lapse of saturation vapor density of 1 gm. me-3 km.+ At time =0, the
atmosphere is everywhere saturated. The heavy lines show the concentration of condensate in gm./m.3 and the light lines show the
amounts of vapor required to saturate the air. The upper row illustrates the case in which condensate evaporates instantly when it
fallsinto air that has become unsaturated by virtue of its history
in the descending branch of the circulation: the lower row illus-
0.5 L/2 L/2
X
FIGURE 6.-Streamlines of condensate (light) and of vapor (heavy
lines corresponding t o those shown in figure 3). The paths of
condensate apply t o the case V/wmoz= -2. The boundary sepa-
rating saturated from unsaturated air is shown for times t=O,
0.45 Hlw,,,,, 0.90 H/wmaz, and 1.35 H/wm.. by the lines of in-
termediate weight. Integers on the streamlines of Condensate
refer t o the steady-state solutions shown in figure 7 .
0.2 1
O*' t 1
JANUARY 1963 MONTHLY WEATHER REVIEW 21
INSTANTANEOUS EVAPORATION OF CONDENSED WATER IN UNSATURATED AIR
I .
0 .
0 .
0.
5 0.
Y w o .
ru 0.
0.
0.
0 .
225 sec. 45-0 sec.
NO EVAPORATION OF CONDENSED WATER I .o
0 .9
0 .8
-0 .7
Y w O .5
E 0 .6
0 .4
0.3
0 .2
0. I
n
675 sec.
0 '0.5 L/2 L/2
FIGURE 5.-Same as figure 4, except that wmaz=
0 0.5 L/2 L/2 0 0.5 L/2 L/2
X V=-lm/sec W max=2m/sec
.2 m./sec. and the time for substantial overturning of the airIis reduced:tol675:sec.
FIGURE 7.-Diagram used t o determine time-dependent distribu-
tions of condensate illustrated in lower row of figure 4. The
numbered lines descending from the upper left corner are the
steady-state M-distributions along the streamlines denoted by
corresponding integers in figure 6. The time intervals occupied
by M-packets descending from one height t o another are given
by the lines ascending from the left.
22 MONTHLY WEATHER REVIEW JAKUAEY 1963
IF 0
FIGURE 8.-Photograph of the time-height recoi-d made with a 1.25-cm. vertically pointing radar. Vertical lines indicate 5-min. time
Wcak generators and trails also occur in the second row; intervals. Precipitation stalactites are most pronounced in the second row.
trails without detectable generators occur in the first row; and strong generators are in the fourth row.
type, wherein the downdraft area is much more widespread
and the downdraft intensities correspondingly more
gentle than the updrafts. The solutions in such cases
can be produced in just the manner already shown. At
the updraft center, the solutions are the same as those
already discussed when the vertical velocity distribution
is the same as that given by equation (20) ; elsewhere, the
principal differences are described in terms of a reduction
of the linear extent of the inflow from the descending
branch of the circulation, decreased linear extent of the
outflow aloft, and somewhat higher precipitation effi-
ciencies in the evaporation cases, since less dry air is
interposed between precipitation aloft and the ground.
I t is obvious that a higher percentage of the condensate
formed in updrafts must reach the ground when updrafts
and conipensatiiig downdrafts are separated by great
distances, and when downdrafts are spread over wider
area and are therefore comparatively weak. It is also
obvious that for a given circulation in an initially unsatu-
rated atmosphere, more of the water condensed in updrafts
will evaporate again before reaching the ground than in
the cases illustrated here. 01 course, many kinds of circu-
lation caii be imagined, and a study incorporating thermo-
dynamic principles quantitatively would be necessary to
determine which ones probably occur.
The method of solution by use of streamlines rather
JANUARY 1968 MONTHLY 'CI.TEATHER REVIEW 23
than a rectangular grid is the more inherently accurate
and can be modified at the cost of added complexity t o
take account of various rates of evaporation. For exam-
ple, the streamline method can be used to study instan-
taneous evaporation by adding values of M >O atid q<O
algebraically after a short time interval to define a new
field of l M f q from which the new locus of M+q=O can
be determined. It seems however, that little would be
gained by introduction of complexities of this kind without
generalization of the assumptions discussed in the
introduction to this study.
6. THE KINEMATIC BASES OF GENERATORS
Generator is the name given to the sources of precipita-
tion streaks (usually snow) observed visually and by
radar; like practically all precipitation sources, generators
are identified with vertical circulation of air. A history
of a generator circulation and the processes affecting it
and its associated trail of snow is incorporated in the water
content distribution. Generators and trails have been
studied extensively by Marshall [lo], Atlas [l], and
Douglas, Gunn, and Marshall [a ]. Weak generators are
shown in figure 8. In this section, the application of
kinematic theory is shown to provide bases for estimating
the lifetime of generators arid the strength of their vertical
currents frorn radar and visual observations.
I n this investigation, it has been assumed that the
generating circulation is given by equations (20), (21),
and (22) with J(z)=O; the environmental wind does not
vary with height in the generating layer. I n figure 9, the
wind shear below the generating level has been taken as
5 m. set.-' km.-' The generating cells are 1 kni. deep and
G, in equation (19), has been taken as 0.7 gin. ITI.-~ km.-',
a representative value for winter near the BOO-mb. level.
Precipitation in the trails falls 1 m./sec., representative of
snow, without growth or diffusion. The water density
distributions within the cells have been scaled from those
illustrated in the upper rows of figures 4 and 5 according
to the discussion given in section 5, and the trails plotbed
in accordance with the equations that relate fall velocity
and wind shear t o position relative to the generators, as
discussed in the references.
With the given assumptions, all the possible shapes of
the M-distributions are implied by the possible range of
the ratio VJW,,~. The ratio - V /W ,~~=~, is given by the
patterns at the left of figure 9, and - V/wmax=1/2 is given
by the patterns a t the right. The labeled w:Lter contents
are for maximuni updrafts of 0.5 and 2 m./sec., respec-
tively. Tn both cases, the circulation starts at time t=O
and proceeds until t= 1.44 N/w,,, when substantial over-
turning has occurred,* and then ceases. In these models,
the precipitation that descends into air that becomes sub-
saturated in the generator downdrafts evaporates instantly.
Many features of the model distributions in figure 9
are similar to the trails observed visually or by radar. ln
both model distributions, the maximum value of M is
4 1.44 Hlwm,,=48 and 12 min., for left and right patterns, respectively.
inside the generator during tlie earlp, active stages and
in or near the generator during most of the period of simple
l'ullout. Second, water content distribut,ions in the trail
of the, generator with weak updrafts are constant or
slowly varying over the trail length, in good qualitative
agreement with observations of inany trails. The short
trails and large condensed water contents of the model
generator on the right are suggestive of manj- observations
thltt have been subjectively- and qualitatively associated
with relatively vigorous convection. Third, the lifetimes
of the model generators confirm our mostly casual observa-
tions that generators seen on radar and cirrus and alto-
cummulus floccules seen visually, are often tracked for
tens of miles and as many minutes.
Study not detailed here shows that the most prominent
features of the water distribution shapes in the left and
right of figure 9 are little different in association with
respectively weaker and stronger updrafts. However,
where -V and toTnaz are about equal, the shapes of the
condensed water distribution are very sensitive to the
value of V/wma2. This indicates that generator and trail
observations might be useCully separated into classes where
updrafts are either less than or greater than the terniinal
lalling speeds of the precipitation involved. A uniform
long trail such as that illustrated on the lel't oE figure 9 is
reasonable only when the fall speed ol' the precipit a t' ion
exceeds the updraft; the frequent occurrence of this type
of observation indicates the prevalence ol' convective circu-
lations in snow with vertical speeds rather less than 1 m./
sec. The same core-water content distributions and con-
clusions are valid if the wind field is radially symmetric or
il the generator dowiidral'ts are more widely dispersed in
a two-dimensional configuration than assunied here.
1Iie solutions suggest inquiry into other quantitative
relationships between the possible observable cell and trail
parameters and the ratio V/wma2. Where the core updraft
profile is parabolic with height and precipitation fall veloc-
ities are uniform as above, and where the air circulation is
steady from t=O to t==CH,'w,,,, the following relatioriship
gives the duration of st,eady precipitation at the midpoint
of the generator base :
H
Wmaz
r i
I
t,=- [C-(--K-l)-'/2 tan-' (--K--l)-''*], (25)
where K is equal to V/w,,, and is less than -1. When
there is simple descent in the region below the generutor,
t , limy be related to the vertical depth D, of the part of a
trail with a steady core intensity by the relation D,=t,V,
and the duration t (of the whole precipitation-releasing
circulation) might, be enipirically related to the total h i 1
depth by the relation D-tV.
The iiumerical value of the steady core-water content
at the generator base and in t8he trail under tlie same
conditions as above is given by
M=ZHK ( -K-1)-1'2 [I - tan-' (-K- 1)-"211, (26)
where -(J is the average value of G in the generator layer.
24 MONTHLY WEATHER REVIEW JANUBRY 1963
L Y k
,111 M Y
k k
JANUARY 1963
--- - __
MONTHLY WEATHER REVIEW 25 . -
The maximum steady water content at the generator
base for a fixed amount of air overturning occurs when K
is such that steady conditions within the generator core
are just attained when the air circulat,ion stops. When
KL -1, there is no steady state; the methods used in
these cases to compute the time-dependent M-distribu-
tions in the updraft cores are discussed in section 1. These
methods have been used to obtain, in figure 10, the dashed
part of the curve which shows the maximum water con-
tents at a point midway between base arid top of the
generator.
7. THE STALACTITE PROBLEM
Stalactites, as observed by vertical-pointing radar,
are illustrated in figure 8. The best published discussion
of stalactites is probably contained in Douglas, Gunn,
and Marshall [2 ]. These radar-observed phenomena
occur when stratiform precipitation descends into dry
air layer, thereby destabilizing it by evaporative cooling.
The depth t o which overturning of the air then occurs
and the extent of the associated downward projections
of precipitation (stalactites) appear to be governed by
such factors as the initial vertical distributions of tem-
perature and moisture and the precipitation rates.
In the stalactite layer, precipitation descends most rapidly
in the downdrafts and is slowed or suspended in the
updrafts. Moisture descends irregularly, and new con-
vective cells probably arise beneath the old as the precipi-
tation lowers. Stalactites are more pronounced in snow
than in rain, in part because evaporation from relatively
fast-falling rain is associated with a smaller vertical
gradient of the cooling rate than snow is. In this section
the application of kinematic theory shows that the sta-
lactite observations may be explained by vertical currents
of about 1 m./sec.
Equations (19), (20), (21), and (22) describe a model
stalactite situation provided that the condition M=
constant is applied a t z=H and the interior of the cir-
culation cell initially contains dry air. The outstanding
feature of this problem is the M-discontinuity that sepa-
rates descending condensate and the dry air beneath.
A proper solution of the instantaneous evaporation case
includes accurate delineation of this boundary. However,
the digital techniques described in [5] are inadequate
in the presence of this discontinuity, and the extension
of the computational method used to obtain the solutions
shown in the lower rows of figures 4 and 5 is too laborious
for hand calculation and has not been programed for a
computer.
A simpler problem is defined by the assumption that
evaporation is very slight (just enough to start the con-
vective cell and to keep it going!) and that the leading edge
of precipitation is therefore given by the locus which con-
nects points descending a t speed (V+w) along stream-
lines of M>O. For the two-dimensional example with
- V=2wmaZ, isochrones of the leading edge starting with
-x-
FIGURE Il.-Horizontally tending lines are isochrones marking the
leading edge of nonevaporating precipitation which descends from
z =H a t t=O. Descent through the wind fieldwhere V/wma,= -2
is along the vertically tending streamlines. Time labels apply if
V= - 1 m./sec., wmoz=0,5 m./sec., and H= 1 km.
t=O at z=N are illustrated in figure 11. An analytical
expression for the stalactite length in this no-evaporation
case is given by V(t,-t,), where t , and td are the times
required for -condensate to traverse the updraft a t x=O
and the downdraft at x=L/2, respectively. Where V
is constant and the vertical w distribution is parabolic,
the stalactite length S is given by
The graph of equation (27) is given in figure 12.
A more probable wind structure associated with sta-
lactites consists of a central core of strong downdrafts
surrounded by a ring of much weaker updrafts. At
some distance from the strong downdraft, the vertical
motion is zero. In this case, the stalactite length is
deduced horn study of precipitation descent in the down-
draft core and in the no-draft region considerably removed
from the core. The applicable equation, whose plot is
also illustrated in figure 12, is the same as equation (27),
except that the term (1/2-) tan -l (l/Y-) is replaced
by 1/K.
i
26 MONTHLY WEATHER REVIEW JA4xu.m~ 1063
FIGURE l3.--The time for precipitation falling at 1 m./sec t o de-
scend through a depth II= l km. is indicated for two downdrafts
(lower curves), two updrafts, arid a quiet atmosphere. I n each
case, precipitation is assumed to evaporate instantly in sub-
saturated air, the initial distribution of M is M= - 24- (z /H )
(gm./m.3), and the upper boundary value is M= 1 gm./m.3
I j =m x
O I O O
FIGURE 12.--The ratio of stalactite length S to cell depth H is
plotted ayaiiist - V/W,,~,,= for cases of 110 evaporation in two kinds
of wind fields discussed in the tcxt.
The no-evaporation plots in figure 12 represent conserv-
ative estimates of stalactite length, in that greater
lengths are suggested by an elementary study of evapora-
tive effects, described below. With this conservatism of
the present theory in mind, consider V=-1 m./sec. in
the radially symmetric case, and note that a circulation
cell with maximum downdrafts of 2 m./sec. would give
stalactites half the depth of the cell. If updrafts are
locally as widespread and as strong as the downdrafts,
the maximum stalactite lengths could be as great as the
cell depths with maximum vertical currents of only
0.5 ni./sec. If the atmospheric circulations in depth are
about the same as their horizontal spacing, then the stalac-
tite observations themselves suggest that vertical drafts
of about 1 m./sec. are all that is required to explain the
observed stalactite lengths. While these analyses of the
generator and stalactite mechanisms by no means prove
that intense vertical drafts do not exist, they do provide
a rational interpretation of the radar observations and of
light turbulence observed from aircralt flying near the
bases or tops of altostratus layers.
This study ol the stalactite mechanism has been ex-
tended by consideration of the instantaneous evaporation
case as it applies along the special lines s=0 and x=L/2,
where there is no horizontal advection. The simple
equations that facilitate solution along these verticals
are based on an extension of reasoning discussed in
sections 1 and 2. Consider first the case of instantly
evaporating precipitation falling into an unsaturated
updraft. The air above tlie level to which the leading edge
of precipitation has descended is saturated and condensn-
tion therein causes the growth of precipitation above that
leading edge. The vertical distribution of precipitation in
the updraft above the precipitation base is, therefore, the
same as that previously computed in the one-dimensional
updralt case except for the additive constant in equation
(3), M(N), the precipitation water content at the upper
cell boundary. Equation (3 ) also defines condensate
distributions in tlie saturated descending air overtaken by
precipitation falling along the line z=0. The t h e
between the initial state and the final steady state in a
small height interval can be calculated lroiii the steady-
state final condition, the initial condition, and the wind
field. The equation used to determine the tiiiie elapsed is
a finite-difference forinulation of equation (1) for the one-
dimensional incompressible case, viz.)
, . , LTY'.
-
The distributions of all quantities in equation (28)
except At are specified, and it is therefore simple to solve
for At. Equation (28) has been solved for At in five
different vertical air currents where the initial moisture
distribution is defined by Mi= (-2 +z /H ) gm./m.3,
6=10-~ gm./m.4, and II=103m. The results are illus-
trated in figure 13, where i t is seen that the pasticular
.TAXU-4RY 1963 MONTHLY WEATHER REVIEW 27
assumptions regarding the rate of addition of moisture a t
z=H and the iiiitial dryness of the air lead to a balance
between the vertical advection of dry air and the descent
of condensate in the updraft cases. In these updraft
cases, therefore, stalactite lengths as measured by the
difference in height of the precipitation base at x =O and
x=L/2 would be indefinitely long. Of course, the cell
that gives rise to the stalactite phenomenon does not have
an indefinitely persisting circulation. If it did, it would
eventually become saturated throughout by a return
flow of vapor from its downdraft portion and the portion
ol‘ the cell in which precipitation is held aloft would
become indefinitely smaller with time. Circulations
actually decay before such limits are reached.
The foregoing leads to another interesting consideration.
The steady precipitation emerging beneath the updrafts
of a convective cell imbedded in a saturated atmosphere
has a higher intensity, and that emerging from the down-
draft side has, due to evaporation in saturated downdrafts,
a smaller intensity than the precipitation entering the cell
a t the top. This indicates how, in a vertical air circula-
tion, stratiform precipitation falling through the circula-
tion is redistributed and emerges with horizontal gradients
of intensity. The distributions for a saturated atmosphere
are easily computed, and may be useful in interpreting
radar records where small convective cells are established
within widespread precipitation by microphysical effects,
as is often the case, For example, near the level where
snow melts to rain (Wexler 1161 and Newell Ill]).
8. CONCLUSIONS
The continuity equations are powerful tools for illu-
minating fundamental properties of wind-water relation-
ships. The greatest advances should come as theory and
observation are combined, so that each supplements and
complements and indicates paths of useful development
for the other. Application of kinematic theory to analyses
of conventional data with satellite photographs and distri-
butions of radar reflectivity and Doppler velocity should
yield improved descriptions of the wind field accompanying
precipitation and of the associations of these fields with
wind shear, static stability, and other quantities of dy-
namical significance.
It is important to generalize the theory to account for
and evaluate the bulk effects of cloud physics processes.
This avenue of study is relevant to the problems of weather.
modification by such means as artificial seeding. A pre-
liminary discussion of such a generalization is contained
in IKessler and Newburg [7].
As dynamical numerical models of the atmosphere
become more sophisticated, equations of continuity for
cloud and precipitation in more nearly natural forms will
be incorporated therein and should reveal how the air
motions, water transports, and cloud physics processes
interact to determine the scales, shapes, and intensities of
coiivective events. The recent works by Li11y [t i ], Malkus
and Witt 191, Ogura and Phillips [12], Saltzman [13], and
Sasaki [14], for example, may provide suitable thermo-
hydrodynamic frameworks Ior such advanced studies.
ACKNOWLEDGMENTS
The manuscript was reviewed by Dr. Edward Newburg
and Mr. George Kern, of TRC, who also contributed via
several stiniulating discussions. Mr. Albert C h e l a , of
the Weather Radar Branch, Air Force Cambridge Re-
search Laboratories, determined the solutions illustrated
in the lower row of figure 4, and Mr. Paul Duchow, of
The Travelers Research Center, Inc., computed those
shown in the lower row ol’ figure 5 . A h . Rudolf Loeser,
of AFCRL, and Mr. Brian Sackett, of TRC, drafted the
figures.
REFERENCES
1. D. Atlas, “The Radar Measurement of Precipitation Growth,”
Sc. I). Thesis, Massachusetts Institute of Technology, 1955,
(See Appendix).
2. R. H. Douglas, R. 1,. S. Gum, and J. S. Marshall, “Pattern
in the Vertical of Snow Generation,” Journal of Meteorology,
vol. 14, No. 2, Apr. 1957, pp. 95-114.
3. E. Kessler, 111, “Radar-Synoptic Analysis of an Intense Winter
Storm,” Geophysical Research Papers, No. 56, U.S. Air Force
Cambridge Research Centcr, Oct. 1957, 218 pp.
4. E. Kessler, 111, “Kinematical Relations Between Wind and
Precipitation Distributions,” Journal of Meteorology, vol. 16,
No. 6, Dec. 1959, pp. 630-637.
5 . E. Kessler, 111, “Kinematical Rclations Between Wind and
Precipitation Distributions, 11,” Journal of Meteorology, vol.
18, KO. 4, Aug. 1961, pp. 510-525.
6. E. Kessler, I11 and D. Atlas, “Model Precipitation Distribu-
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Tropical Precipitation and Kinematic Cloud Models,” First
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039 sc 89099, Travelers Research Center, Inc., 1962.
8. D. K. Lilly, “On the Numerical Simulation of Bouyant Con-
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Element; A Numerical Calculation,” in The Atmosphere and
the Sea in Motion, Rockefeller Institute Press, New Pork,
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11. R. Newell. ‘.Some Radar Observations of Tropospheric Cellular
Convection,” Research Report No. 33 by Weather Radar
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tract AF 19(604)-2291, 1959.
12. 1’. Ogura and N. A. Phillips, “Scale Analysis of Deep and
Shallow- Convectioii in the Atmosphere,” Journal of the
iltmospheric Sciences, vol. 19, NO. 2, Mar. 1962, pp. 173-179.
13. B. Saltzman, “Finite Amplitude Free Convection as an Initial
Value Problem, I,” Journal of the Atmospheric Sciences,
v01. 19, NO. 4, July 1962, pp. 329-341.
14. Y. Sasaki, “Effects of Condensation, Evaporation and Rainfall
on the Development of Mesoscale Ilisturbances: A Numeri-
cal Experiment,” Scientific Report No. 1 , Texas A. and M.
College on Contract AF 19(604)-6136, 1960.
15. A. F. Spilhaus, “Raindrop Size, Shape and Falling Speed,”
Journal of Meteorology, vol. 5, NO. 3, June 1948, pp. 108-110.
16. R. Wexler, “Ad\ ection and the Melting Layer,” Meteorologzcal
Radar Studies No. 4, Blue Hill Meteorological Observatory,
Harvard University, Contract A F 19(604)-950, 1957, 12 pp.
pp. 36-40.
1959, pp, 425-439.