) a cumulus
ence theory_
cumuliform.
l. J.,
1953:
J. Meteor.,
Penetrative Downdrauqhts
tribution of
ofizicheskaia
in
Cumr-rii
By P. SQUIRIS, C.S.LR.O., Radiophysicr"r"Oorrrory,
(Manuscript received October
r,
Sydnev, Australia
1957)
Abstract
observations of cumuri have shown that the adiabatic rnoder is
dry air must nrix with the condensing Llpcurrenf. Theories of the quite inadequate, and that
interaction båtween cum*li
and their environment are discussed.-Th! hypothesis it"t d.y
air enters these clouds chiefly
from above is examined quantitatively. It is found thar, as a
of evaporative cooring,
1erylg
such air could penetrate severar kilomeires into a gro*iog
croud. This rryporÅ.ri,
for
the fine structure which is obsewed, and providå rotrrrr and
"..ounts
simpie explanation
for the
fact that the laps,e rate in curnuli is steepei than the" ;;; ;JJ;;.,';"'j-r"JJ.i'"rr.oximates
closely to that of the environment. Unlike arternative theories,
t; ;;;;r;
;."t"", *rrrr.uy
the liquid water content may be automatically selfJimiting,
as the o6servations ,..* ,, ,.qu,..,
for the motions which introduce drl'air deåp into the cioud dep.nJ f"r-,r*ir-""..gy
,"ppru
of liquid water, and have i-elocities .hi.h
,;-ir;
"..';;;s^ht'f."p".,i",."r
:å:::,.n"f:."1.e
r.
Introduction
A wealth of evidence has been accumulared
which shows rhat the properties of cumuh
cannot b.e explained by t6e aåiabatic expansion
ot arr which was saturared at cloud base level.
Measurements of temperature in cloud describ_
ed by Buxxrn et al. (r9a9) and by Banrurr and
krur
(1948) indicate thar the'averase lapse
rate in cumuli is steeper than rhe *.c åirb.'ti.
and_approximates to that of rhe environment.
Additional evidence is afforded by measure_
ments of liquid water contenr (,, e m-r).
Afcer reviewing the observarioni, fXzan*rn
a,nd Sgqrrp (1958) concluded that although
rne adlabaltc llqurd water contenc (øo) some
tlmes occurs over a distance of the order of a
metre,- the average value of u, ovet some
tens of merres (rø) seldom if rever equals ru^.
The ratio Dfwo
and at
(
(1ej8), 3
Tellus
X
deueases
z km above
(1958), 3
rapidly wiih heiefir
cioud- base falls belåw
o.2, on the_ average. There is iittle evidencc
to suggest that rrr reaches m-rrch greater value,
tn large.ctouds than rn-moderate ones; in ail
cloucls,
tt
m-3, and
As
seems rare
it is
lor rz to
in th. horironirl,
a great wealrh of fine stnrcture,
regards distribudor
rh,ele, rs.
which
: e
r e m-i
exceed
ustrally less than
becomes very srriking when rø is
measured over a discance of ibout r metre.
-iråå
There is. no ciear systemaric
"rri"tior,
the ciou.d edges towards the middle.
Conrrary
to rhe rmpresslon given by Ztrusnv (rSsoi,
tne low values of l, appear to be distributed
rather randon-rJy, with'no marked
pr.f.;;;.;
tor the penpheral regions; dre edge bf a cloud
ts olten verv sharD.
r;These. nnaings' clea-rly require for their
.lplTitto,n
a modei of cumulus developmenc
whlch tncludes as a necessary concomitånt of
a
large siate
:1. wlth
:-1..:lil-growrh
alr
the upcurrent.
*i"i"t;f ;;
P. SQUIRES
, Q1
Jw2
z. Theoties of interaction between the cloud
and its envrronmenr
Since cumulus bases are typically flat, ir
seems impossible
for dry air !o enter the cloud
from belåw, for in the-present context, "dty"
air means air which, if displaced adiabatically
to the level of the cloud base, would arrive
there unsaturated. The variations in cloud
base height, which rarely.exceed one_ or two
hundreJmetres, are much too sma1l to explain the observed deficiency of liquid
-air water,
which
i*" - tr.). Clearly, however, all
åråir-tri.r from above the cloud base level
wifl be "dry" in rhe sensc defined. Thus there
is room foi chree possibilities:
dry air may not, have moved, far
- (a) The
rts orlgrnal posltlon, navlng rruxed. ln
trom
situ with theiaturated, cloudy airlising from
belov.'.
6) It mav have come in from around the
sidls of the'cloud, mixing horizontally with
the updraught.
/^\ T- r1r4/,
from the regior.
'-^,- L^..rrdvv vrrt
^.;oinared
. \L,/ ,t.
r
Just above the gro\'lng,clouoitopJ enterlng
rhe cloud as downdratlghts ancl mlxlng vertically with the updraught.
-telated
to the bubbie
tr,todel (a) is clbsely
theory of convection advanced by Sconrn
and Luoralur (tSl:), according to which
the erowth of a cumulus is the result of the
,tri.r- I at condensation level of a succession
of buoyant bubbles, each of which is steadily
eroded'as it rises. If other bubbles follow
the same Dath throush the air, they will
Denetrate fårth.r b.."tise the ambient air has
6een moistened, by, thgr5 predecessors; by this
orocess a cumulus butlds up m steps.
^ Mod.l (b)was postulatediy Sror"t'ær (r9a7)
to explain ihe obse*.d lapse rates in cumuli,
and h"s been used as a basis for discussion by
Marxus ft949, t95z). The latter author has
hacl considerable success in interpreting observations by means of a postulated entrainment.
However, in tåese caies, there has been no
independent evidence relating to the amount
of årainment, and it seeås possible that
some other mechanism which introduced dry
air into the cloud in the required amounts
could explain the observauioni equally
-o[ well.
entrainIndeed- tle ottlv direct evidence
ment is orovided bv measurements of wind
fields aroirnd laree iumuli and thunderstorms
I
made during the Thunderstorm ProjectBylns and Huru (r9a9) have described how
the proportional rate of change in area of
trianilei formed bv three balloons were used
the
is
tl
assig
,"
clou
to comp,tte horizontal divergence. The typical
even
rate oi entrammen! lnto tnese targe clouos wa$
found to be roo % in + km. By contrast,
Srorvrurr (loc. cit.) computed that, in order
to explain'the obierved iapse rates in small
trade wind cumuli, it was necessary to postulate an entrainment rate of ioo % in a height
interval o[ 3oo to r,ooo m. Again, BEsr (r95r)
M
/;)
cum
envi
æ
mea
drau
warl
in large
of ait of 7o61o
humidirv at a rate similar to that
computed the liquid water contenr
level
.,-rrn'oli assuminE'entrainment
exist
relative
found by Bvtns'and Hurr, and found values
chan
and
exist
of ø which far exceed those which have
usually been observed in iarge cumuli; the
by
corresponding values o[ vf wo exceed o.5 au
all heishis. Ii therefore seems that, if the en-
mea:
SOm(
trainm-ent rates measured by Bvlns and Hurr
are typical of cumuli in general, this source
of div air by itself is inJdequate to explain
th. otserved properties of such clouds.
cum
regir
ålser'
^hai
been outlined elsewhere
(Squrn-rs, r958) in an rttemPr to e-xplain some
o[ the observed characteristics of cumuli. A
parcel of dry air wlLich enters the top of a
cloud bv turbLrlent diffusion wlll be
åto*it',e"by
the evaporation of liquid water'
fooled
and mav iubside into the ciotrd. The existence
of a -åchanism of this kind is suggested b,v
measurements of vertical velocity, such as
those oIJoNrs (r95aa), which have frequentlv
rcvealed an jtrtimate mixcure ol up-and*downdraughts of the same order of magnitudeAeain, the downward motions often seen
a.6und the outsides of cumuli in lapse time
records are commonly attributed to åvaporative cooLins consequen! on mixing between
irt div environment.-I[ this exthe cioud
"id
it seems likely that the
is
correct,
olanation
ii-il"t mechanism of model (c) åou1d lead
to deep penetration of a cloud by dry air
Model
(c)
and
t9-l
ant
clou,
frorr
into
In
ance
taker
hydr
clou.
that
mass
the
evaP
efan(
menl
unstt
clou,
(ii,
parcelsl This model, Like (a) and (b), explains
åuafirativelv the observation thai the effects
of the admixture of dry air increase upwards
through the cloud, as shown-by the decrease
of. wll.u-. It sives a picture of cloud development ;r *iik. thåt of model (a). In boih
cases, the final result is a heterogeneous mixture of ai.r from below condåsation level
with air which originally occupied, roughly,
Tellus
X
(1958),
l
t
i
1&.
wa,
I
over
ffi t
:4ft=
ffi
;{t.
mad,
I
4
291
that
mecl
of dr
ones,
I ellus
PENETRATIVE DOWNDRAUGHTS IN CUMULI
roJect.
how
rea of
e used
rypical
os was.
.ii
.&l
å-t
order
small
Postu-
height
(rs s r)
'1";*:
,..670%
) that
ch$g.e must take place berween the cloud top
values
have
i; the
and the clear air-above
existence
Hun
xplain
where
some
,ill
be
from the evaporation'of cloui p.otob..".r..,
into the surrounding dry air. -
yr/ater,
itence
it
iikely that the protuberances at the,rop of.growing cumufi:rnay be
raKen as evldence of the exlstence of strons
hydrostatic instability in the resion of th?
cioud top; if so, there is little r."ått to doubt
that downdraughrs are entering the cloud
mass with a vigour similar to th;t shown by
the protuber"ni.r. Buc whereas cooline by
evapårariontendsto stabfize rhe cloud pr8roterances when they mix with the drv eivironment, it renders' the dry air parcels more
unstabie as they mix with the'surrounding
Indeed,
:d bv
rs
rentiy
iownitudeseen
time
Pora:ween
is exLt the
lead
seems
cloud.
Y, air
(ii) As mentioned in the Inroducti.on, rhe
of liquid water content so far
made indicate that] wharever rhe vaiue of
rua, w usually averages less than r g m-3
over large voiom.s, .id ir
to lxceed
"ttlik.ly
2, g m-,3 except in smali regions.
This suggests
that there rs some torm of self-limrtinq
plalns
measurements
:ffects
wards
:rease
'elopboth
mixlevel
ghlv,
eJ8),3
"who
and stood in strong c-ontrast ro the undistuibej
temperature fie1d iormally found in the clear
air above cloud base and well awav from
clouds. These cold regions may have resuited
,ofa
.h
(r'qs+)
measured the air remperature while' fying
some r50 m directiy a6ove small developinø
cumuius ciouds,
foo.rd th"t th.r. å.rE
regions about roo"r,å
m across in whiåh the temperature was some r.5o to 3' C colder than
elsewhere; these regions were sharp-edged,
iource
A
ir. Ividence of the
of rhis kind of interchanqe is given
by the observations of Javrs
o.5 at
Ji.
now {illed by cloud. The difference
penetrative role is
assigned to the dry _air. Thus a heterogeneous
cloud with much fine structure coulå result
even from a sready updrauqht.
Model (c) has ce.t"ir advåncaqes:
(i) The remarkable facr thar -lapse races in
cumuli do not differ much from ^thar of the
environment finds a simple explanation by
means of the mechanism oT penet?"tive downdraughts. If the cioud should^become markedly
warmer than the environment at the same
level, an extremely unstable situation will
exist at the rop of'the cloud; vertical interthe
-space
here. an active,
is that
I
t
i
mechanism at work, that is, that the admixturE
of dry air is greater in wet clouds than in dry
ones. In both the bubble model (a) and thl
Tellus X (19J8), 3
383
enrrainment model (b), the admixture of drv
air is not directiy afiecied by rhe concenrrario;
of liquid water j in model '(c), it is the liquid
water itseif which provides'ihe drivine fårce
for the motions wlich give rise ro åi"it o
As will be seen later, thE subsidine r;;;ffii
dry air parcels in cioud is roughly påpårtional
Eo w.
(iii) It is a common observarion that turbulence as observed from an aircraft is much
stronger in clouds than in the clear air between
them; according ro Buuxrn er al. (r9a9) by
at ieasr one order of magnitude. Now it is
clear that the turbulence asiociated with clouds
derives its er,'ergy from density differences, so
that the turbulint morions Åust in the first
place be verrical; cloud turbulence is likely
to be far from isotropic, vertical mixing ps6dominating srrongly iver horizoncrl *Yås
(iv) As å.otioo".å rn the Introdiii"" ir*i.
is no clear evidence for a gradient of proper-
ties from the cloud centr-e or.rt towaids'th.
edges, as musr be expected under model (b).
The horizontal increåse of. w on entering 'a
cloud is often very rapid, and there is "no
counterpart here to-the irertical sradient of il/
which Wanrqrn (rsss) found to oicur regularly
in the uppermosr 3oo- 5oo m of cumul]. This
contrast reinforces the argument of (iii) above
indicating thar vertical
rnanr role.
The hypothesis
"*1"-g
pl"ys
'" ;;;:
is therefore advanced that
the mechinism of model (c) consrirures an
important aspect of the interacti.on betweerr
-their
cumuli and
environment. It remains
now to investigate this model in a quantitative
manner, that is, to determine w-irether the
air in the dry iayers which the growine cloud
penetrates is'irsålf capable of p"enetrati"nu the
cloud mass in a manner which-is unorsanized
on the scale of the cioud. This penJtration
yy be envisaged as occurring hy n: agency
or more or less con[muous downdraughcs,
for which there is some evidence (loils),
I9j4 (b) and Wanurn, rgjJ), or in rhi form
of individual parcels. The latter view has been
chosen in the interests of theoreticai simplicitv.
Å penetrating dry parcel will obvio.usly be
suDJect to attrltlon and may å! trmes be corn
Thus a laree paråel may enter the
"sunder.
.1"."a, penetrate. sdme^ distance,' and rhen
dlslntegrate, glung flse to two or more
smalei on.s .irhi.li thereafter follow a more
P. S QUIRES
384
or iess independent coutse. As our PurPose ls
merely to investigate whether the ProPosed
mechånis* is capil'le o[ introducrng dry air
into the interior of a cloud- ln a tlme mterval
comparable with the observed lifetime of
.o*'uli, the subsiding parcels will
be supposed
to femaln mlegral.
3. A physical model of the downdraught
theoty of interaction
The following Postulates,describethe Pllsical
modei whtch ls used tor tne purPose or comDUtatlon:
'
(a)
,l
parcel of dry air becomes immersed
in ihe cloitd top as a result of the.erratic growth
procuberances, or by turbuJence
oI cloud ^by
the increased hydrostatic inreinforced
stability which must tend to occur at the
cloud iop when the lapse rate of,the environment is steeper than the wet adlabatlc, as ls
usually the case.
'tvlixing occurs between the parcel and
(b)
its clor.rdv envitonment.
(c) Cloud droplets {itrupg into the parcel
bv' iurbulence immediareiy-evaporate- and,
the parcel is cooled. Cloud droplets
",.rnlt, in tinsaturated air, even at a rela"i
immersed
tive humidity of 99 %, wili evaPorate to a
diameter of a micron or so in some tens of
seconds, unless they contain
a.
giant condensa-
nucleus; these are relatively rare. Thus
tion
thev will have lost practically a1l their mass
witLin a time which is short'compared with
the periods consideredfdi fhe resistance to the motion of the
p"i.ål thtoogh the cloud arises soleiy from
åddy
inr...hi,g.
of marter between the parcel
and'its enviroiment. The disturbance of the
pressure freld in the environment consequent
Lpott th. motion of the parcel is neglected.
fhis simplification is justified to sorne extent
by che.foilowing consideration. No hyqothesis
is'made as to the shape of the parcels,,and
thev mav be envisaged as quasi-spherrcal or,
mo're plausibly, as bJing elongatedin a,verticåL
direction, so that they torm short-llvcd downdraushts- In this case, the effect of the disturbthe ptessure fie1d could well be small
"n.."of
compared idth thar of eddy
interchange
bet#een the parcel and the cloud.
(e) The paicel is envisaged as containing a
.oåri",,, *"ts o[ air. Following, the method
used by PntEsrrrv (1953) it will be assumed
that, in unit time, a mass o[ air k is removed
from each unit mass of air in the parcel and
is lost to the surrounding cloudl this is replaced by an equal m"tt df cloudy air, which
inixes with the-parcei. The effect o[ the parcel
on the cloud will be neglected in this calculation: obviously however the final effect of
many such parcels will be to cool and dry
rhe åloud. especially in its upper parts.
The mixins rate L mentionå in-(e) depends
on the effectii'eness of eddy diffusion in transferring matter from the cloud- to the parcel
,r-ice versa. For the moment it suilices to
"nd
remark that k varies inverseiy with parcel
size; the relation becween them will be
thr
m(
thi
IJ
m:
rvi
as
on
tht
tio
m(
tht
€Vt
m(
discussed later.
In addition to these postulates, the following
simpliil.cations and asi,lmptions are made:
rhe- density difrerence
(å) In
Sn
ag
_computing
b.iween the pårcel and its environment, temoeratures,
.toi ..ittu"l temperatures, are
used'
'sinc. the parcel itself is clbse to saturation in
the criticai stages of its hisrory. the neglect
of the effect of humidity mixing ratio on the
air densifv is not very serious' Moreover, thrs
simolfica'tion results'in an underestimate of
the'negarive buoyancy o[ the parcel' and so
reduces the comptLted dorvnward sPeeds: 1t
sp(
'r
OI
ha'
t
co_
Pr{
cu
4.
is therefore conseivative.
(b) The comPutarion begins. at, a point
wherc the parcel ts tmmersed ln the clouc:
it is at the'same temPerattlre as the c1oud,
and is at rest.
(c) The liquid water content of the cloud
ir å[..t tt b.-g constant throughout.its depth'
fd) The cloud air is assumed to have unlform and constanl upward velocity'
(e) The lapse rate^in the cloud is taken as
wl
Pal
tht
wl
Pa
constan!.
(f) The mixing ratio of water vaPour 1m
the cloud air is fotrnd assumlng the alr to De
,"*."t.d and at a temPerature determined
by (e) above.
'R'l/ th.se postulates seem reasonable for the
purpose in hand excePt perhaps (c)'. This rs
åulf. .otot.v to th. ådi"b"tic-model, which
steady decrease downwards to
zero at the cloud base; however, it represertts
the observations at least as well ås any other
simple assumPtion'
curves o[ the
\*/^o*uo (i955) shows
åould indicaie
thr
a
-typical
*lrt-t-tt.igl'tr ."f .å-. peak,value o[
""tittlo"
liquid water content ('ar) tottd,u:Ii3r,:i
A
Pa
ler
th,
of
tht
tul
ser
th,
iiq
thr
Tel
7-
1
PENETRATIVE DOWNDRAUGHTS IN CUMULI
I
aircraft traverse of a cioud. These curves indicate a rapid increase from zero at cloud base
through th. fitrt 5oo to rooo m, and a rather
more rapid decrease at cloud top. Berween
< is removed
Le parcei and
1' fhrc rs rF-
ly air, which
of the parcel
this calcula-
L
nal efect of
;ool and dry
Parrs.
n (e) depends
in trans.^
.v ,LLrrL y4rLLr
-^-^^t
sion
it
sufiices to
."irL
^"..^l
em wiil
be
åe following
are made:
-.r
.]
iffo"o.""
)nment, tema"a
"a"
,,""1
saturation in
*L- *^^l--.
I Llru raLårLLL
'ratio on the
loreover, this
erestimate of
arcei, and so
..1
....1c'
ir
at a Point
n the cloud;
rs the cloud,
of the cloud
rwo regions, the height variirton of wo
is differenc in different clJuds. and no svste-
?
matic average trend is apparent. The behaviour
che meån value o[ p, (r),
from the peak value (ru"), depends
on the shape of lrheirr-traces obiained during
-W-a.nNrn
(private communicathe traverses.
tion) states that in about half the clouds
measured, D behaved similarly to wD; in
the remainder the height variation of i was
even more erratic than thar of l,/.. Measurements by Day and Muncarndyo (rps:),
'WucruaNu and AuFM Kaupl (1953) and
Squnrs (1958) give indications which are in
with heighl o[
as distinct
agreement
"As
have been carried out for only one set of
conditions, which have been chosen to represent a fairly well developed trade wind
cumulus.
4. Basic equations for an unsaturated parcel
On the basis of the postulates described in
the last section, and ntglecting some terms
which are small because the density of the
parcel is little different from thar of the cloud,
ihe equation of morion of the parcei is: ,, .,
is taken
as
err+vapour in
the a1r to De
:
determined
for the
s (c). This is
rnable
nodel, which
lownwards to
,
it
represents
as
any otner
curves of the
,eak value of
rd during
an
Iellus X (19J8), l
The humidicy mixing ratio o[ the parcel
to the iqu"ciorr; ,'l " '. ^, .
varies according
i :
where
r
(:)
is the mixing ratio of the parcei air
th"'cloud air at rhe level z.
^F
(r)
Differentiaring
gives:
..Å
i:6s-ki,
A fto^ (z) and lor /
from (r) qives a second order linear differendal equådån in u. Taking as boundary condi-
and substituting for
tions y : o, A :c at t : o, and defining the origin
of z, measured downwards, as the position
^t.1". -"..^l ^i
r-^ .te following expressions
result:
bt
:6" ,-u,
tuc - tng
A-
sirr
år
+ 6"f
-L,
- s-"
(r
cos
*& f{ . ry\l
J \rø" tn/ J
år)
(+)
-NI
-:-t-ffi
LUc
(r
\(Y*
- ru,cos år) + lg.
Lmry ffi,\tuc
_
l/åt
where
i\:
|
=J
tnz
The liquid water content of the cloud (w")
enters into this equation because the suspended
Iiquid water. increases rhe effecrive densiry of
the cloud arr.
X (19J8), 3
7-80JB't I
- x)
k(x,(z) + w,
rn.l
4rrs . /-) rhrr
^c
the level concerned.
Tellus
I .
[,,_:..
the iapse rate in the cloud.
-(r)r"'
. la
\ g-k.-.(u+ I/) '(j'.:
i:t+-ru.l
\9" "/
where y is the downward velocity of the
parcel, Z the speed o[ the cloud upcurrent,
/ the amount in degrees C by which the
oarcel is colder than the cloud at the same ievel, @" the meen absolute remperature o[
the cloud, and u" rhe liquid warer conrent
of the cloud in gråms p.r'grr* of air. Since
the overall variation of the absolute temperature of the cloud is only a few perceni, no
serious error is involved in usins the mean
temperature @" instead of th. t.rip.rature at
1r
I
where I is the latent heat of evaporation of
water, 14 the.dry adiabatic lapse iate and l"
these conclusions.
so many variables are'involved in the
speci{ication of the problem, and the solution
of the equations is laborious, the calculations
to have unicrry.
with
- {r,'' r"), ø
: p(+ - t\
/
\r,
Å
these
rout its depth.
Ld
38j
.The equation governing the temperature
ot the parcei rs:
i ol 7
å
e_kt sin
skL ,
=-Occp
"
r.-l
:, "..
t" : _ u.i-
"l'i
it
1
/
br
k'V
w,
vc
4-
m
:f ! rr,- ,r^)l't'
",1
LO"
and m: kz + !a
j.
ry)
(ro - r").
(s)
l,
"
3
P. SQUIRES
B6
TABLE I
of the downward motion of dry air parcels
through a cloud against a steady uPcurrent of r rn sec-r'
Characteristics
Liquid water
content oI
cloud (g/kg)
*"1
r.o
03
Mixing Rate,
to-3 x
Final R. H. %..
99.r
"Absolute" Pene-
tration (-)....
Time Taken
3r
8re
secs.
"Reiative" Pene-
tration(-)....
rr 42
26o rR<r
,
1
98.5
roo n" ul
3ooolr5T3
866 .r-ool
738
o.37
o32
t
I
3ooo
-c 36oo 2999 2466lrri
Final Tempera-
ture Deficit ('C)
96.7
^"_t^^-
3ooo
;:"1,;:.
r9831 roo4
435
5
91
l""l
5999
54661262
t67 4ft3ot oro"l 498j1
o.40
o41
o.6:l o.83
r.27
z-6t
o.861
o.8rl
r25
75
598
4oo-1
37 5
3598
24
o.9
I. 4
7'70
3.4
Final Downward
Velocity
(m sec-r) ....
1
i. e. To cloud
o.77 o.5
.
3
base
Usine rhe valtres of .z as e Function of r given
by.(s), ." -"y be found numerically or grrphically from rhe solution of (3) :
Consequently, the point where saruracion
firsr reached'is takcn as tl're terminus.
The numerical eval-rations have been carried
or-rt
xoe -kt
+ ke
kt
a
|
,ot
(r"
(z) + w")
dL
(o)
rs is the original nixing ratio of
the
Darcel ai;. Saturatiån is reach-ed when the
iralue of r found from (6) equals the saturation
mixing ratio of air at the remPcrxttrre xnd
Dressure
^ When
ot tne Darcei.
the paicel becomes saturated, it will
continue to tubtid. for a time becar-rse of its
temDerature deficit and downward momenturr-ti Cons"quently, some of the liquid water
diffusine in must continue to evsporate rl
the oaråel air is to remain såturated. In these
.i.c,jr.rrtanc.s the modified equations corresponding to (r),. (z) and (:), ibove must be
tieated as simultaneous. The equation to
which they iead, however, is non-linear, - so
become more complex
that the computations
^case
of the unsaturated parcel.
than in tåe
will be seen, the penetration of unsaiurated
parcels is sufiicient t6 estabiish the quantitative
ådequacy of the downdraught hypothesisAs
its
Thus 6,:2$s"I<. l':
per km.'The parcel enrering the cloud at
6'C
t:o 6 assLlmed to tave an initial
relative humidiry of io qo. With a Pressure
corrcspondinq to 4 km (ICAN) and a remperature o[ ]z'C, this gives .16 : :.68 g/kg'
The speed of the updraught, V, is taken as r m
-liquid \,vater content
sec-l. Three valuås of
are used, o.3, r.o and 3.o [7kg. rhe first two
velues may reasonably be,taken as rePresenring
average condtttons ln dry ancl wet clouos
,.rp.åirr.lv ('Wanrqnn and' Squrnrs 1oc. cit.) ;
it approximately the mean of
the'last
"rluåvalue^of this cloud, which varies
the adiabatic
from zero at rhe base to 6.2 g/kg al the toP'
the niornent
å,
where
for a cloud based at r km (16'C) with
top at + k- (-z'C).
t
x:
is
lt
has been included
in
order
to
iliustrate
what rapid and deep penetrations would
occur in'a cloud of *.fi high liquid warer
content. The vigotrr with whicli dry air
parcels would peietrate it may expiain why
iuch hish val.,ås of tu have so fai been observed ånlv in isolated regions of cumuli.
Calculatåns have been iarried out on the
basis of equations (+), (s) and (o). The results
'1'ellrrs
X
(19J8)'
3
PENETRATIVE DOWNDRAUGHTS IN CUMULI
I
of interest are shown in Table I. The calcula-
I
I
tions have been terminated either when the
parcel saturates, or when it subsides 3,ooo
I
ttt I
,.f
i-.--i...-----l-
;11."
4.o
|
I
5.o
I
nn.,
roo
1'lOOo
720
598
r30
j
6.0
"l
IOO
I
I
I
I
Lz5
250 245
3598
3.r4 3.42
5.r
I
metres and so reaches cloud base, or after
the lapse of an hour, which ever occurs firsr.
The '?absolute" penetration in Table I is the
final depth of the parcei beiow its startino
point. 'heladve" påetration is the depth o?
Zloud above ir ar ihe end of the calcul'ation,
assuming thar rhe cloud top continues to rise
at the sfeed o[ the updrauqht aft.r the oarcel
enters tlhe cloud; sirice
_dri air parceli
may
enter the top of a cloud ei1ual1y well at any
stage of grdwth this figure gives an idea o'f
t"r"t
irow
deeply they may become buried in the
cloud as å result'of their own subsidence, and
the cloud's growth. For each value of the
cloud liquid -water conrent, on-1v those vaiues
of k ar{ shown which are r..årr".t to illustrate the behaviour of parcels of varying
size. As will be seen from åquation (z) beiow,
the parcel size varies inversåly wirh k. Smail
parcels (k > 6 x ro-3) are 'scronslv cooled
ind
;aturaclon
1s
Ilus.
been carried
'C) with
r8o"K,
its
I":
the cloud at
ve an initial
:I alPressure
ano a tem= 2.68 g/kg.
tm
rter content
he first two
:aken as
rePresenting
wet clouds
:s l^. .it \'
he mean of
vhich varies
^f
rlta
+^h
to lllustrate
Lons would
iquid water
cn dry
a1r
xplain why
rr'been ob-
f
cumuli.
out on the
lne resurts
Llus
X
(1958), 3
develop hiqh åownward #..dr. buc
saturate quicklyl, Iarger p".c.ls^ (,å < ro-s)
remain unsaturated for
a
long time, but, except
w :-: e/ke, they aie
only slightly cooled and subiidJ relaiively
slowly through rhe updraughr.
rt'l
Y'
Lable Ir rndrcates
that parceis of a suitable
size can penerrate deeply into srowins cumuii
in a period .o--..lå."t. *i"th the iife-time
of thise clouds, especially when ir is remembered that they ian enrer at all stases of
growrh ("Relarive" penetration). The åesrec
6f cooling and the'falJinq
speed are #oth
roughly lroporcional to "rhe^ liquid water
content of the cloud, and the valuei flound for
them are consistent with observed fluctuations
in cloud, especially in the more realisric cases
Luc:o.3 and r.o g/kg.
An inspection of Table I wiil disclose some
apparent anomaiies - small changes in å sometimes give rise to iarge changes in parcel
behavidur. These disco-ntinuitie! are dile to
the fact that both rhe temperature deficit of
in the unrealistic
case
the parcel and its downwarå velocity have an
exponentially decreasinq sinusoidal .ornoor..rt
(Equations (+)
and (s))l fhe rejative ]iumidt''
.t
t
1
1\J/1
rry' ofr the
parcel shows a similar behaviour.
For certain values of the parameters, the parcel
approaches saturation q-uite closelv ."il,r itt
iti^history-one of rhe^peaks of rhe relåtive
Tellus
X
(19-t8), 3
387
humidity curve reaches closc to roo % -but
nevertheless it remains unsatrlrated for a
considerable time afterwards. In thcse circLlmstances, a small change in the parameter
k may have the result th-at the peal exceeds
roo o/o, that is, the parcel saturatås. This kind
of discontinuity in -the data of Table I does
noc necessarily imply an anomalv in rhe phvsical behaviour of'the parcels. It'is due å ihe
-of
arbitrary termination
the calculations ar
the point where saturation is first reached.
TlruJ it seems likeiy that in a case like LUc : 3
g/kg, å : 5 x ro-3, where saruration is jusr
reached at rhe firsr peak of rhe relative hrrmidiry curve, the patlel would in facr quickly
dry oul agam and contlnue clownwards tor
some distance before finally satllrating.
""
Ir ic itr nnc.ihle
oirza lllutc 'l'".
tllalt a "-.'
\ cry
-._ tn
"_ xrvs
-^..
of rlie size of chesc parceli.
llowever, it can be established that the optimal parcel size deduced from the calculatibns
is not absurd in the physical conrext beins
rotrgh indicarion
the'eidy difftisivity, K, ai
is easily shown that
discussed. Treating
constant,
it
k:c KlR2
(z)
where c is a constant dependine on the forrrr
assumcd for the proFrle'oF pråperries across
the parcel, and R is the paicel-"radius"; in
the lasc oI a verrically eiongrted parcel, R
would be the redius of its Eorizorical cross
section. Thus, in any one cloud, Å may be
regarded *, m."r.r.. of parcel size, ani chis
rernrins valid" evcn rhotrsh-rhe eddv diffLrsivirv
depends on the scale of ihe ph.noåeno.r, thJt
is, the effective vahre of Kreillv depends on R.
Pnmsrry (1oc. cit.) has discussed tjre value of
the constant c and has concluded that c - 8
is a reasonabie choice. The correct value of c
is noc likely ro deparr from this by more
than a facror o[ z. ilowever, rhere såems to
bc no way oI arrivinq ar an appropriate value
"arr
'mrsnitude.
for K even within
ordei'of
The eddy difrusivitv which is effeJtive in
catrsing mixinq between the parcel and rhe
clot'd Is rhac die to eddies sienihcantlv srnaller
than the parcel. Ir wili be m-"uch iess ihan that
which might be deduced, for e-xampie, from
an aircraft accelerometer record, assuming
the turbulence to isotropic. for this woulå
include che effects of rhe åotion oI the parcels
themselves. On the other hand. it wili evilentlv
exceed the sma.ll-scale backsround turbulencå
P. SQUIRES
388
of the clear air outside the cloud, since the
motion of the parcels, and of the upcurrent,
will sive rise to additional small-scale turbulence] It seems in any case unlikely that the
^*-*^-*;^'^ -'^1"dyyLvyLLaLw
v4ruL
^r K should iie outside
vr
the range ro2 to 106 cmz sec-1. Taking å :
z.ro-3 ai the optimai value, the resuldn{value
of R from (z) ls found to be in the ranqe 6 to
6oo m. Thus'the optimal parcel sizes dEduced
from equation (7)-are plausible in relation to
!ii
?i,
Sit
irl
åi
the phenomena under discussion.
table r indicates that, for any one value of
rhe effective eddy diffusivity, only a narrow
range oI parcel sizes are. capable of,achieving
appreclable penetratlons ln a reasonabie perlod
of time. In actual clouds, the steady state
assumed in deriving rhis result never occurs,
and moreover the parcels must often be
reduced in size by attrition or by disruption.
a l^-^^ ^"..^l '-'^,, ^-r.-.!,- -l^,,1 ^";1,,;ics growth and, although incapble"of subsiding
tar, may later acl as a source ot dry alr tor
the resion near cloud base. Thus no iiteral
sisnifiånce can be attached to the detailed
vilues in Table r. The calculations can be
regarded only as showing that certain downr"
t
t e'
1
draughts (ot a physically plausible sizc) are
.
of achievinq siqnificanc penetretions,
so'that the proposed itteÅ.nit- foi inroducing
dry air inco a cloud is quanritatively adequate.
There is, of course, nothinq in the calculations
themselves to indicate holi often such downcapable
in
clouds, but the physicai
argumenrs oF Section : indicate that they are
draughts occur
common.
5.
,"1
iil
;i
i:!:
-"il'
Conclusions
Theories of the interaction between cumuli
and their environment have been reviewed in
the light of the existing observations. Physical
argllments have been given which indicate
rhåt the region around ih. rop of a growing
cloud may be an imporrant source oI the dry
air which mixes with the upcurrent. A theore-
that parcels of drv air can remain unsaturated
whilå subsidins up to three kilometres (i.e. to
cloud base) thiough the updraught in å time
which is co--.niur"te with rhå life-time of
cumuli. Even after saturating, such parcels,
-
being colder than the surr-oundinq cloud,
*ooi"id continue to subside for some- time as
resions of relativelv iow liquid water content.
"A 1".e. nu-b.i of sucir parcels enterins
the ciouå top ac all stages of iti growth woulå
result in a wåalth of firie strrrct.rå in the distri-
bution of liquid water content and other
cloud propertiis, and would have the efrect of
reduciiE
the liquid water content and of
cooling" the ctoud, especially in ics upper
p"tm. Thur the hypothesis affords an expiåna-
iion of the presence o[ fine strucrure
It
also
'explains
{
4
the upward decrease of the
T
*
ratio wlw], and the friquent occurrence of a
cloud lapse rate steeper than the wet adiabatic.
On this view, tG initial immersion of dry
air parcels in the cloud is grearly accentuated
*
ii
whån the cloud rop beåornei appreciably
warmer than the environment. ln addition,
the deeoer Denetration which follows is
achieved-by åotions which depend for their
liquid water,
energy,
lupply on rhe presence oI
and whrch rt appears have velocltles roughly
proportionai to it. Thus a very wet cloud
would be more susceptible to deep penetration
than a drv one. The mechanism described
cherefore prouides a type of autornatic action
which noi only ensrrr-ei that the lepse rare in
the cioud shali not depart far from that o[
the environment, but aiso hmits the concentration of iiquid water. Both these aspects
of the hypothesis accord well with the observat10ns.
A corollary of the hypothesis is that there
should be a radical difference in structure
between convection in the clear air and in
cloud; the presence of the liquid phase in
cloud introduces an element of-instabilitv bv
dry downdraushts has been set uo, and the
caiculations ."åed out for a cloud i.k* d..p,
with an upcurrent of r m sec-1. As can 5e
providing a source of energy which is available
to drive"smaller scale -oiionr. This situation
has no analogue in the case of clear air convec-
seen from^ Table
uon.
these calculations show
,ii
"nd
the fact thatihe observed liquid water content
is nearlv alwavs less than the adiabatic vaiue.
tical model of the penetraiion of a cloud by
r.
l
of
H
Æ
B
,id|
4
w
q
#
.;:
,.*E:
"tellus
X (19i8),3
'l:'
CUIVlULI
PENETRATIVE DOWNDRAUGHTS IN
389
ated
:. to
REFERENCES
:ime
eo[
."1.
.e
+
as
tent.
rin-bo
---
ouid
temperature
;;t,; H., t94g: Vertical distribution of
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,"T'it-rariv'M.r"ot.,
and woods
Mass' Inst' Tech'
Hole Ocean. Inst. XI No r'
patterns.of
Bvsns, H. R., and Huu, E' C', r949: Inflow
"^?;;;";t;".ms
as shown by winds aroft' Bull' Am'
)Ieteor. Sot 3o,90-96'
tli-:l^lt
D,rv, G. J., and Muncarnovo' R' J, r953:,The
cioud investlgatlons mrde by the Mereorological
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Research Flight during the period
r952. M.R.P. 826.
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a: Five flights through a ThunderTo*es. R. F., r9s4
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Ivlet. Off., London, 7, No rog'
6..-e"^']-;;;
rr of
Io[
PPCT
ana-
dof
IlenI
a1ue.
' the
ofa
)atic.
: drv
lated
;^1-l-,
ir
5, 30+-307.
of ice
Brsr, A. C., i95r: The occurrence of high rates
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A. F., Haunwrrz, B', Merrus' J S and SrolrBuNxrn,
-
"
tstri,ther
nn
's
'W', and Rtsnr, H, r948: Experiment-ai veriE.
Meteor''
Iication of entrainment o[ air into cumtlts' J '
BARRET,
arra]
is
their
ratef,
rohlv
-J-_-/
:1or-rd
at1()n
ribed
ction
.te in
at of
lCeniPects
,bser-
there
.cture
rd in
se in
:v by
4
rlaDIe
ration
nvec-
eJ8),
3
'IelLus
X (19J8)'
3
Marxus,
J .s: I?1?"_tu:;':,,?' ;;:!. å,',;;."tt,:'[:
of convecllon'
1
aspects
oI convcctivc
iii''ia*..'It rdvances in che scudy
- .io',At
thcir interxction with the envlronment'
Ttlltts, "tta
4, 7t-87.
a turbuC' H. ts., r953: Buoyant motion in
Pnrrsrrrv,
' "'i.n,
6'
P^ltv1'.tuirotmenr' '4trsr' J'
:1?. i?"-'
i9)J
'.,,*^'.
S.onen, R. S., :nd Luoi 'ttur' r' n"
Qttnrt' J' xoy'
conrecLion
theory of pcnctrrrire
Md.
Sor.
sq"tJt,"l.,
96-103'
79'
'i;rs,
The
variation
?t,t]ni.1-.Yii
,spatial
concentrations ln cunull
ter content and droPlet
Tclftls. ro, J7: -J8o
--..r.,-
a cumulus
sto*"**, H.l,"tgilt'Entrainment of air into
c1oud. J. Meteor. 4' 9t-9+'
r
1:c--*
tntt'The rveter content ot curnulllorm
w;;,i.,
clotrd. Tellus 7' ++9-+57'
rnd S'quiott, P'' r958: LiCuid wrrelr-o.1t31t
.t.d th'c rdiabaric model of crtmtLlus develoPrnent'
Icllrrs. ro, 39o- l9+'
W^--tt.l'
wrr"*ti"*,-i.
r., and Kautr' H J' Auru'-1953-t\4cfe or
clouds' J'
The physical propertles of cumulus
lo,20+-21r'
content and-distriZ^t;;il"+. Å., ,qlo' Licluid rvater clouds'
Lcningr"d'
bution of drops tn tt-t'lu'
r9
al;":"y" Geofråcheskaia observrtori)'' I rudy'
(8r): r:z-r32.