Met Office College - Course Notes
Cloud physics
Contents
1. The development of cloud physics
2. Aerosol spectra
3. Droplet formation
4. Rate of growth by condensation
5. Growth by collision-coalescence
6. Ice processes
7. Drop size distributions
8. Cloud seeding
9. Lightning – cloud charging mechanisms
10. References and further reading
 Crown Copyright. Permission to quote from this document must be obtained from The
Principal, Met Office College, FitzRoy Road, Exeter, Devon. EX1 3PB. UK.
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“So the cloudlets grow by mutual fusion and scud before the winds,
until the time comes when a raging storm arises”.
Lucretius 1st century BC
1. The development of cloud physics
Cloud-seeding studies in the USA and Germany in the late 1930’s and
1940’s highlighted the dearth of understanding of cloud precipitation
mechanisms. By the 1960’s cloud physics laboratories were investigating
basic ‘cloud physics’ The developing requirements of Numerical Weather
Prediction intensified the need to understand, and parametrise, cloud
physics processes. Studies have been increasingly complemented by
fieldwork with instrumented aircraft, radar and aided by satellite
imagery.
1.1 The cloud-modelling problem
The atmosphere produces precipitation, both a sink of moisture in the
atmosphere and a source at the surface. An NWP model must remove
moisture from its grid in order to simulate this process. The scheme ‘has
knowledge’ of temperature, water vapour, liquid and ice content centred
in each grid box. The model must estimate changes in these quantities
due to precipitation processes. There are many physical processes
important in clouds (see selection below); the rôle of the ice-phase is
being increasingly appreciated.
Table 1 Some important cloud physics processes/properties
Droplets
Ice phase
Hail
Raindrops
Activation
Nucleation
Dry hail growth
Melting
Condensation
Deposition –
growth,
sublimation
Wet hail growth
Collision with ice
or droplets
Fall speed
Collision &
coalescence
Spectrum
evolution
Droplet shedding
Shape
Fall speed and
mode
Ventilation
Collision,
aggregation
Collection of
droplets (riming)
Splintering,
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Nucleation of
supercooled
drops
Evaporation of
raindrops
Fall speed,
ventilation
Drop break-up
For all processes:
The influence of
Cloud physics
multiplication
electric fields
Parametrisation means processes must be simplified – the scheme must
capture the necessary physics from a knowledge of only a few variables.
A suitable ‘scheme’ is shown in Fig. 1. Thus a cloud droplet is assumed
to respond instantaneously to changes in the model background
humidity – the fine details of the Kelvin, Solute and Köhler curves are
irrelevant! (Wilson and Ballard, 1999).
Processes leading to precipitation formation
Fig. 1
Parametrisation needs an in-depth appreciation of the processes listed in
Table 1 and Fig.1 to establish which ones are less/more important for
the modeller. There are still many important and exciting discoveries to
be made in cloud physics, from the evolution of the droplet spectrum to
the nature of cloud charging mechanisms. Some of these problems have
been occupying cloud physicists for over half a century!
2. Aerosol spectra
Aitken in Scoltand and Coulier in France, working in the 1880’s,
established the fundamental rôle of condensation nuclei for cloud
droplet development; Aitken worked on the nature of aerosols in
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different air masses for over 30 years. Fig.2 contrasts the cloud droplet
spectra in continental and maritime air masses.
The German investigator, Junge, working in the 1950s, found that
particles of r > 0.1m obeyed a size distribution law of the form:
n(r ) 
dN
A

d (log r ) r 
where A is a constant and dN = n(r) d (log r) is the number of particles
per cm-3 in the radius interval (log r). Furthermore Junge found  to be
typically about 3. This implies that the sum of the cubes of radii for
particles in each logarithmic interval of radius is equal. In other words
the contribution to the total aerosol mass from particles between 0.1 and
1m radius is similar to that from particles of between 1m and 10m
Junge: n( r ) 
dN   A
d log r 

r
Junge : n ( r ) 
A
dN

d log r 

r
dN  n ( r ) d (log r ) : particles cm
3
in radius interval :d (log r )
• ~3 so  r3 in each log r interval is equal
• Mass contribution 0.1 to 1 similar to that for1 to 10 }
• 0.1 to 1: accumulation mode; 10 to 20 coarse particle mode
• encouraged by Brownian aggregation; and by gravitational settling
radius (Fig 3).
Fig.2
Fig.3
Fig. 3 shows size distributions of aerosols collected over Miles City,
Montana, the ‘envelopes’ containing many different samples. As well as
particle number, there are distributions based on surface area: dS/d(log d)
and volume: dV/d(logd).
The Junge distribution, which would appear as a straight line in Fig. 3,
illustration (a), only provides a rough fit to the data which show distinct
inflexion points. Furthermore, if  = 3 provided a good fit to these data,
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the trace in Fig. 3(c) would be nearly horizontal. The maximum between
0.1 and 1m, especially noticeable in Figs.3(b) and 3(c), is known as the
accumulation mode, which is explained by the tendency of particles
smaller than 0.lm to collide with one another due to Brownian motion
and coagulate. Between 10 and 20m exists another maximum known as
the coarse particle mode, which is dominated by dust, combustion
products and sea salt. This has been shown to depend on wind speed
and distance from the sources of the aerosols. The reason for its
occurrence is the tendency for gravitational settling to remove larger
particles.
It is interesting to note that it is the large aerosols (0.1 – 1.0m) which
contribute most to the total aerosol surface area.
3. Droplet formation
3.1 The adiabatic cloud liquid water content.
A first approximation to the liquid water content of a sample of cloudy
air can be made by assuming adiabatic ascent of moist air. The mass of
liquid water is given by the initial humidity mixing ratio of the air
parcel, (rso), minus the current saturation humidity mixing ratio, (rs),
taking account of cooling at the Dry Adiabatic Lapse Rate while the air
is unsaturated and Saturated ALR while the air is saturated:
q l = rso - rs
This calculation can be made on a thermodynamic diagram, such as a
tephigram, and converted into liquid water per unit volume rather than
unit mass by multiplying by air density, or p/ RT. In practice, however,
actual values of cloud liquid water content are lower than the adiabatic
value, suggesting that mixing of drier air from outside the cloud occurs
to a significant degree.
3.2 The Kelvin (curvature) effect
The equilibrium vapour pressure over a curved surface of water is
greater than that over a plane surface. This can be explained in terms of
surface tension or in terms of molecular attraction. At a curved surface
there is a weaker net attraction holding water molecules in the liquid
mass, since each molecule at the surface is more exposed. Consequently
more molecules escape into the vapour phase than over a plane surface
and so the vapour pressure exerted is greater. This pressure is given by:
 2 

e s (r )  e s () exp 

 rR v  L T 
…(1)
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where e s() is the equilibrium vapour pressure over a plane surface of
water,  is the surface tension, r the radius of curvature, Rv the gas
constant for water vapour and L the density of water.
As a droplet's size increases, the vapour pressure exerted decreases, and
liquids with higher values of surface tension (and therefore greater
molecular attraction) show a proportionally greater increase in vapour
pressure over a curved surface. While e > e s(r), a droplet will grow by
diffusion of water vapour, i.e. condensation, whereas it will shrink by
evaporation if e <e s(r). A critical radius, r c may be defined for a given
vapour pressure by writing
e = e s(r) in equation (1).This defines the
size of a droplet that will remain in equilibrium for a given vapour
pressure. Setting e/e s() = S, a measure of relative humidity:
rc 
2
Rv  LT ln S
Thus high supersaturations are required for very small droplets to
remain stable. For S = 1.01 (i.e. 1% supersaturation), r c = 0.121m at 00C,
while for
spontaneous (homogeneous) nucleation S must be very large (~ 6 to 8)
since chance aggregations of water molecules form only tiny droplets.
3.3 The solute effect
Molecules of solute at the surface of a droplet reduce the number of
escaping water molecules by occupying sites on the water surface, thus
lowering the vapour pressure. For a plane water surface, the reduction
in vapour pressure due to solute is given by:
n0
e'
n

 1
e s () n  n 0
n0
(2)
where e' is the equilibrium vapour pressure over the solution, n 0 is the
number of molecules of water and n is the number of molecules of
solute (n<<n 0). If the molecules are dissociated into ions, as is the case
with a salt, n must be multiplied by a factor, i, which represents the
degree of ionic dissociation.
i = 2 is considered appropriate for
many calculations. Introducing a constant, b, equation (2) simplifies to
(Mason, 1971):
e s' (r )
b
 1 3
e s ()
r
Combining this result with Kelvin's equation (1),
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e s' ( r ) 
b 
 1   exp
e s ( )  r 3 
a
r
where: a = 2 / R vLT
Providing r is not too small, a good approximation to this equation for
realistic values of a and b is:
S
e s' (r )
a b
 1  3
e s ()
r r
(3)
3.4 The Köhler curve
To find the maximum value of S in equation (3), in other words the
greatest value of equilibrium relative humidity for a droplet of given
solute concentration, dS/dr = 0, i.e. - a/r2 + 3b/r4 = 0. This gives:
r* 
3b
a
Substituting into the expression for relative humidity:
S*  1 
4a 3
27 b
So the maximum equilibrium relative humidity is greater than 100%, the
margin of supersaturation depending on the temperature, density,
• Equilibrium saturation ratio of a solution droplet formed on
Ammonium Sulphate nucleus, mass 10-18 kg.
surface tension (factor a), and the solution concentration (factor b)
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Fig. 4
Fig. 4 shows a Köhler curve for a water droplet growing on a salt
nucleus of ammonium sulphate of mass 10-18 kg.
When the radius is small the solute effect is important, while there is a
fairly rapid transfer in dominance to the curvature effect. In general, at
droplet radii above ~5m, neither effect is significant, i.e. the
equilibrium relative humidity approaches 100%, (S=1). Droplets of r < r*
are termed haze droplets, remaining in stable equilibrium for a given
RH, whilst droplets passing r* in radius become unstable and will
continue to grow, reducing the environmental supersaturation and
causing the haze droplets to shrink. Table 2 shows examples of critical
supersaturations for droplets formed on sodium chloride.
Table 2: Values of r* and S* as functions of NaCl nucleus concentrations at
273oK
Mass of dissolved salt
(kg)
r* (m)
S*-1 (%)
10-18
0.0223
0.42
10-17
0.0479
0.13
10-16
0.103
0.042
10-15
0.223
0.013
10-14
0.479
0.0042
3.5 "Cloud processing" of aerosols droplets;
The Met Office’s Met Research Flight results (Osborne, 1996) suggest
that various processes, acting during the condensation/evaporation
cycle of cloud droplets, can result in residual aerosols which are highly
effective as cloudy condensation nuclei , CCN, Figs. 5 and 6. Processes
are such that:
a) aerosol mass within a droplet can be increased by absorbing SO2.
Ionisation and oxidation leads to sulphate ion formation and, ultimately,
a larger more effective CCN;
b) aerosols too small to be CCN initially can be scavenged to increase the
aerosol mass;
c) coalescence of cloud drops also results in larger residual potential
CCN material;
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d) entrainment from above the cloud layer can result in a local, sparser,
population of larger drops;
e) dimethylsulphide (DMS) from the air-sea interface is thought to form
CCN by homogeneous nucleation. Thus increases in DMS are thought to
Ship’s plume aerosol size distributions,
AVHRR imagery 27/6/87, western seaboard of USA.
Ships’ tracks in low level cloud.
Fig.5
• High levels of anthropogenic CCN are provided to Sc-topped boundary layer.
Aerosol spectra, measured by C-130, within plume near cloud base.
• Pronounced bulge after 120km with local max at 0.3m - due to ‘cloud
processing’. These aerosols likely to be good CCN and effective at scattering
solar radiation (Osborne 1996).
yield greater cloud albedoes through increased CCN concentrations.
Fig. 5
Fig.6
The effect on solar scattering characteristics of the cloud (susceptibility)
have implications for parametrisation, climate change estimates etc.
3.5.1 Cloud susceptibility
Cloud susceptibility describes the sensitivity of cloud top albedo, A,
over the visible wavelength band, to changes in cloud droplet
concentration, Nd (Osborne, 1996):
Susceptibility, S 
dA
dN d
S can be related to aerosol concentration; studies have concentrated on
Sc cloud sheets. The implications of possible anthropogenic aerosol
impact on cloud droplet concentration, and hence on the radiation
balance of Sc layers (and thus on climate modelling), are clear.
4. Rate of growth by condensation
Consider a water drop of mass m and radius r growing by slow
diffusion of water vapour. A steady state is achieved with the water
vapour flux independent of time and distance, R, from the centre of the
droplet. This flux is given by Fick's law of diffusion:
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F

dm
 4R 2 D
dt
R
where D is the molecular diffusion coefficient for water vapour in air
and  is vapour density such that e = RvT. Integrating this equation
from the surface of the drop out to infinity leads to:
dm
 4rD (     r )
dt
where  is simply the environmental vapour density. It can be seen that
if     r the drop will grow, whereas if     r it will shrink. The
problem now is to establish  r' which depends on droplet size, chemical
composition and temperature. In turn the temperature is determined in
part by the exchange of latent heat which must therefore be considered.
In the case of condensation the heat release keeps the droplet
temperature higher than that of the environment. The diffusion of heat
away from the droplet can be treated in a way analogous to the diffusion
L
dm
 4rK (Tr  T )
dt
of water vapour to the droplet, giving the equation:
where L is the latent heat of vaporisation of water and K is the
coefficient of thermal conductivity of the air. Note: the transfer of heat is
from the droplet to the environment. The drop temperature and the
vapour density at its surface are related by the equation of state for
water vapour combined with the formula for equilibrium vapour
pressure over a curved solute surface:
r 
e s' (r )  a b  e s (Tr )
 1   
Rv Tr  r r 3  Rv Tr
No analytical solution exists for this equation, but it may be solved
numerically for temperature and vapour pressure at the droplet surface
to obtain the rate of growth by condensation (Mason, 1971).
r
dr
S 1

  .......... .......... .......... .......... .......... .....( 4)
dt FK  FD
Here S is the relative humidity (S - 1 is the supersaturation). Notice as r
increases, the slower the growth rate becomes. FK represents a
thermodynamic term associated with heat conduction, whilst FD is a
water vapour diffusion term. The greater the thermal conductivity (K) of
the air, the smaller is FK and therefore the faster the rate of droplet
growth. Similarly, the greater the water vapour diffusion coefficient (D),
the smaller FD and the faster the rate of droplet growth. Including the
Kelvin and solute effects:
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dr
r

dt
a b

r r3
.......... .......... .......... .......... .......... ...( 5)
FK  FD
( S  1) 
Again, it is not possible to integrate this equation analytically to obtain
drop size as a function of time, though numerical methods can be
employed.
Table 3 gives growth rate results for droplets growing on nuclei of
sodium chloride at 0.05% supersaturation, p = 900hPa, T = 2730K.
Table 3 Time (secs) for droplets of initial radius 0.75m to grow by
condensation to stated radius (Mason, 1971).
Nuclear mass
(kg NaCl)
10-16
10-15
10-14
1
2.4
0.15
0.013
2
130
7
0.61
4
1000
320
62
10
2700
1800
870
20
8500
7400
5900
30
17500
16000
14500
50
44500
43500
41500
Radius (m)
4.1 Narrowing of the droplet spectrum
To summarise: a droplet forming on an initially large condensation
nucleus grows faster but, after a certain radius (about 10m in Table 3),
growth rates become similar since the a/r and b/r3 terms are no longer
important. The expression (4) for droplet growth rdr/dt can be
integrated:
r (t )  r02  2t
Consider two different sized droplets r2 and r1.
r2 (t )  r1 (t ) 
r2 (0) 2  r1 (0) 2
.......... .......... .......... ........( 6)
r2 (t )  r1 (t )
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Thus for two droplets growing in the same conditions of temperature
and humidity, the difference between their radii decreases,i.e. the spectrum
of droplet sizes becomes narrower (but see Section 5.4). Of course
equation (6) can also describe the rate at which a droplet shrinks in an
environment of relative subsaturation. Using (6) with expressions for
terminal velocity of droplets (which increases with r2 for drops r <
50m), it can be shown that the distance of fall for complete evaporation
increases with r 4. Some results are presented in Table 4.
Table 4 Distance fallen by drop before evaporating. Assumptions: isothermal atmosphere
with T=2800K and RH=80% (Supersaturation, S=0.8). (From Rogers & Yau, 1989).
Initial radius (m)
Distance fallen
1
2 m
3
170 m
10
2.1 cm
30
1.69 m
100
208 m
150
1.05 km
Though results vary with RH, and the expression for terminal velocity
used is not accurate for the smaller droplets, these figures show that a
droplet has to achieve a radius of order 100m in order to survive the
fall from cloud base to ground. Accordingly 100m (0.1mm) is often
taken as the dividing line between cloud droplets and precipitation
particles - the drizzle drop.
4.2 Growth by condensation of droplet populations
Equation (5) describes the growth of a single droplet for given values of
temperature and humidity. In order to extend this to the growth of a
population of droplets of varying sizes it is necessary to recognise that
temperature and humidity will change with time and to consider the
water vapour budget of the cloud volume. The rate of change of relative
humidity can be written:
d
dS
dz
 Q1
 Q2
dT
dt
dt
where dz/dt is the vertical wind speed and d/dt is the rate of
condensation. Q1 represents the increase in saturation through
adiabatic cooling on ascent and Q2 represents the decrease in
supersaturation due to condensation onto cloud droplets. Together with
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equation 4 this equation can be used to examine the evolution of a
droplet spectrum, having specified an updraught velocity and an initial
droplet distribution (e.g. Mordy, 1959).
Typically, droplets growing on the smallest nuclei grow initially but fail
to reach the critical radius (r*) and shrink back as haze droplets under
the solute effect as supersaturation decreases. The other droplets
continue to grow, and the spread in size becomes narrower with time, a
property of the growth equation demonstrated earlier.
Measurements taken with developing cumulus clouds do indicate
narrow drop size distributions centred on a radius in the range 5 to
10m, the average radius increasing with height above cloud base in
accordance with the theory of growth by condensation. However, it is
observed that higher in the cloud, and in the later stages of
development, the spectrum of drop sizes is broader than that predicted by the
formulae for growth by condensation - clearly other processes are at work
(see Section 5.4).
5. Growth by collision-coalescence
5.1 Collision efficiency: theory and laboratory experiments
The series of Figures (7a-d) summarises: (7a) the nature of collision
efficiencies between droplets; (7b) theoretical and experiment data on
collision efficiencies; drops must achieve a size > ~ 20m before growth
by collision can begin; (7c) later studies, which indicate the possible
influence of wake capture or ‘micro-turbulence’ in enhancing collision
efficiencies as r/R approaches 1.
Fig. 7d shows actual collisions
between large droplets in the laboratory (Woods, 1965; Mason, 1971).
Fig.7a
Fig.7b
Collision efficiency, E
R
r
• At grazing incidence:
E
 y o2
 (R  r)2
yo
• Collection efficiency=collision efficiency x coalescence
efficiency (often put=1)
• Collision efficiency, E, for various collector drop
sizes as a function of r/R.
• Theory & experimental data. E
0 for r<<R and r/R ~1
5.1.1 Collision efficiency and drop size spread
•
One of the main goals of cloud physicists has been to explain how
raindrops could be created in as short a time as 20 minutes from
nucleation, a figure often quoted as the observed period between initial
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formation of a cumulus cloud and production of rain. As seen already,
growth by condensation proceeds quickly at first but slows down with
increasing droplet size and it is accepted that there is no way that
precipitation may be produced by the condensation process alone in the
lifetime of a cloud.
Growth by collision-coalescence requires not only the presence of cloud
droplets of radius > ~20m, but also a sufficiently broad spectrum of
sizes. As seen, growth by condensation, theoretically anyway, makes the
spectrum narrower with time. So even if particles grew to 20m by
condensation, according to the simple theory there would not be enough
• Laboratory Cloud Physics
• Computed collision
efficiencies, E, for
pairs of drops as a
function of r/R
• A ‘streak’ photograph
of the coalescence of
two drops, r=62m.
• Curves labelled by R
• Fall speed, hence
drop size, from
‘strobe’ illumination.
• Notice ‘wake’ capture
effects at r/R~1
Coalescence event
• (Woods & Mason 1965)
Fig.7c
Fig.7d
of a spread of sizes for collisions to occur effectively.
Various suggestions have been proposed to explain this discrepancy,
many involving the entrainment of dry air from the cloud top (Telford
and Chai, 1980). See Sections 5.2 and 5.4.
5.2 Growth by continuous collision
Consider a droplet radius, R, falling at terminal velocity, u(R) through a
population of smaller drops radius, r, terminal velocity u(r). In unit time
the average number of droplets collected is:
(R+r)2 {u(R) – u(r)} n(r)E(R,r) dr
where n(r)dr is the number of droplets per unit volume with radii in the
range r to r + dr, E is the collection efficiency (i e. collision efficiency x
coalescence efficiency). Integrating this expression to obtain overall
increase in droplet volume and then manipulating to obtain dr/dz and
dr/dt:
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dR EM u ( R)

dz 4  L U  u ( R)
Then assuming droplets radius, r, are sufficiently small, that u(r) = 0 and
R + r =R, and that the updraught, U, is negligibly small:
dR
EM

dz
4 L
and
dR
EM

u ( R)
dt
4 L
Here M is the cloud liquid water content (i.e. the mass of cloud water
held in drops having radius in the range 0 to R) and E is an average
collection efficiency. These are known as the continuous collision
equations.
Fig. 8 shows calculated growth rates and trajectories of drops, R. While
predicting realistic relationships between strength of updraught, cloud
height and precipitation size, the results suggest an unrealistically long
time to produce precipitation ( >1 hour).
Fig.5
Fig. 8 • Calculations of time of droplet growth to
precipitation for two updraught speeds.
• Assume: collector 20m, droplets 10 m and lwc of 1gm-3
Fig. 8
5.3 Stochastic growth
The continuous collision models take no account of statistical
fluctuations in droplet concentration, dealing in average values only. A
related limitation of earlier analyses was the distinction made between
the collector drop and the collected drops. A different approach is to
consider the evolution of the droplet spectrum as a whole, rather than
classifying a subset of it, as collectors from the start. A coagulation
coefficient, K(R, r) may be defined, representing the likelihood that a
drop of radius R will overtake, and collide with, a droplet radius r,
provided that they are present in unit concentration. Letting V and v
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denote volumes corresponding to drop radii R and r, then let H(V,v)
represent the probability that a drop of volume V will collect a drop of
volume v:
1
1

 3V  3  3v  3 

H (V , v)  K 
,
 4   4  


The evolution of the droplet concentration of drops with volume v is
determined by two processes - first of all a sink due to coalescences
involving drops of volume v to v+dv, and secondly a source due to
coalescences between pairs of droplets whose volumes sum to v.
So the rate of change of drop concentration in size interval dv is given
by:
v

0
0


1
dv H ( , n)n( )du  n(v)dv H (V , v)n(V )dV .......... .......... ....( 7)
2
where  = v - u, in other words  and u are two volumes whose
combined volume is v. The factor ½ is to correct for the fact that the
integral counts each capture combination twice. This stochastic
coalescence equation was actually first derived by Smoluchowski in
1916!
Fig. 9 shows the result of a modelling experiment based on equation 7,
starting from an assumed droplet size distribution at t = 0.
Initiation of rain in non-freezing clouds
30
• Example of the computed development of a
droplet spectrum by stochastic coalescence.
Fig.9
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5.4 Condensation during coalescence
Continued growth by condensation enhances growth by coalescence.
Despite the fact that growth by condensation decreases the difference in
size between drops, the relative fall speeds do not alter, since the
terminal velocity of droplets for r < 40m is proportional to r2 and it has
been shown that the difference between the squares of the radii remains
constant (equation (6)). Condensation increases the average drop size so
average collection efficiency increases, and this effect serves to enhance
growth by coalescence (Fig. 10).
Effect of condensation on growth by coalescence
•
The same droplet spectrum is shown evolving
a) without,
Fig. 10and b) with, an allowance for condensation
5.5 Cloud droplet spectra – the reality
Recent studies, while not invalidating the described stochastic and
entrainment/mixing etc. processes, do indicate that the development of
the required spread in drop sizes is not fully accounted for by such
models. The growth rates observed in the atmosphere are greater;
indeed the theoretical narrowing of the droplet size spread with time is
not observed either.
Evidence points to very small-scale vortices enhancing droplet velocities (and
hence droplet collision rates) beyond those that could be expected in still air.
6. Ice processes
Current research on mixed-phase clouds (Wilson & Ballard, 1999) points
to the crucial importance of the ice phase in precipitation development,
as summarised in Section 1.1.
It must be emphasised that the potential ice nuclei are far fewer than
potential cloud condensation nuclei - perhaps one particle in 10 million
(1 in 107) qualifies as an ice nucleus at - 200C. Fletcher (1962) estimated
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that active ice nuclei are typically present in concentrations of about 1
per litre at - 200C, the number active increasing by a factor of 10 for each
additional 40C cooling. That the number of ice precipitation particles can
exceed these concentrations may be put down to the various ice
multiplication processes discussed in Section 1.1.
6.1 Deposition
The equation for growth by deposition of vapour (Bergeron-Findeisen
process, Fig.11) is similar to that derived for growth by condensation,
being given by:
dm
 4CD(     r )
dt
where C replaces r in the condensation equation, and is a function of
size and shape of the particle. For a sphere, C = r and the equation is the
same as that for growth by condensation onto a water droplet. For a
circular disk of radius r, which can be used to approximate plate-type
crystals, C = 2r/, whereas ice needles may be approximated by the
formula for a prolate spheroid. The deposition growth equation may be
derived in a similar way as that for condensation where r is replaced by
C, es(T) is replaced by ei(T) and L by Ls (latent heat of sublimation).
The equivalent curvature effects are not well understood, and the
equation takes no account of the fact that vapour molecules cannot
deposit onto ice in a haphazard way - they join in an orderly fashion,
molecule by molecule, to maintain the crystal pattern. In addition, the
complex shapes of ice crystals mean that the vapour pressure exerted
may vary across the ice surface. Experiments suggest that at
temperatures between 0 and - 100C small crystals grow about half as
slowly as predicted by the depositional growth equation.
However, the most important difference between growth by
condensation and growth by deposition is the fact that (Si - 1) in the
atmosphere is much greater than (S - 1) where water droplets are
present, by perhaps 2 orders of magnitude, making growth by
deposition a much more efficient mechanism for precipitation
production than growth by condensation (Fig.11).
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Direct observation of the Bergeron
Findeisen process
Ice crystal
Evaporating
supercooled
water droplets
Fig.11
6.2 Ice nucleation mechanisms
Ice may form directly from the vapour phase on suitable deposition
nuclei. Three modes of activation are recognised for freezing nuclei (Fig.
12):
- some serve first as centres for condensation, then as freezing nuclei.
- some promote freezing on contact with supercooled water drops
- others cause freezing after becoming embedded in a droplet.
A particle may nucleate in different ways - a function of history and
ambient conditions; recent work by MRF suggests contact is the
Ice nucleation mechanisms
Hetergeneous
deposition
Condensation
followed by
freezing
Contact
Immersion
Fig.12
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important mode.
The relative importance of the different freezing modes has not been
established.
When in a supersaturated environment it is difficult to distinguish
between deposition and freezing nucleation when ice nucleates on an
insoluble surface. Even when below water saturation nucleation need
not imply deposition because nuclei may contain soluble components.
6.3 Accretion and aggregation
Formulae similar to those describing coalescence may be applied to the
accretion and aggregation processes, but with less accuracy since the fall
speeds of the particles are difficult to specify, depending on their
composition (e.g. graupel, snow flakes, clear ice). These different forms
are in turn dependent on conditions of temperature, humidity and cloud
liquid/ice water content, so the problem is complex (Figs. 13 and 14).
Mass and size of different forms of
ice crystal
Characteristic forms
of ice crystals at
various temperatures
• Ice crystals of differing
shapes, growing on a
filament in a diffusion
cloud chamber with
controlled temperature
gradient
• Mason B J, 1962
• The mass/size relationship is expressed by
empirical formulae of the form:
m=aDb
D: the major linear dimension of the crystal.
• Values of a and b for typical crystal forms are,
for D in cm, m in gm:
• Crystal type
a
b
Graupel
6.5x10-2
3
Thin hexagonal plate 1.9x10-2
3
Stellar crystal
9.4x10-4
2
Planar dendrite
3.8x10-4
2
Needle
2.9x10-5
1
Fig. 13
Fig. 14
6.4 Ice phase versus coalescence
An approximate idea of the difference between the rate of precipitation
initiation by the ice-crystal process and by coalescence can be gained by
comparing the early growth history of an ice crystal with that of a large
cloud drop (Rogers & Yau, 1989) (Fig. 15).
The ice crystal grows relatively quickly by diffusion, surpassing the
initially more massive drop at 75 seconds. After 7 minutes the drop’s
collection efficiency is then no longer small and it grows faster than the
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crystal. The drop reaches ice crystal mass at 30 minutes (r=160m – a
drizzle drop).
It has been seen in 6.1 that because of the reduced numbers of ice nuclei
compared with droplet nuclei, each ice particle can grow to a large size
and achieve a significant vertical velocity, providing a substantial
downward flux of moisture. Water drops, however, must spend time
coalescing into larger drops before they can remove significant amounts
20 mins
10 mins
Ice crystal mass catches
up after 7 mins
• Times required for an ice crystal and a water drop to
grow to a given mass.
• Top scale: drop radius. Dashed curves: rate of fractional
increase of mass - scale on right
of water from the cloud.
Fig.15
6.4.1 Mixed phase clouds- and aircraft icing
Cloud with tops between 0oC and –4oC are likely to consist entirely of
supercooled water drops, leading to aircraft icing problems (Forecasters’
Reference Book, Chapter 2.9.1). With cloud tops of –10oC there is a 50%
probability of ice in the cloud; at –20oC the probability increases to 95%.
6.5 Frontal cloud physics
Frontal cloud consists predominantly of water in its ice phase. Fig. 16
shows the increasing fraction of ice as temperatures in frontal cloud fall
below about –5oC.
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Fig.16
Proportion of liquid water
Fraction of liquid water in frontal cloud
More liquid
More ice
Temperature (0C)
About 1/3 of ice crystals grow by deposition of vapour, about 2/3 by
aggregation.
Size distributions of ice crystals, constructed from particle image data,
obtained during studies by the Meteorological Research Flight, are bimodal, suggesting aggregation of ice crystals as they fall through a cloud.
7. Drop size distributions
For steady rain, intensity R, at continental mid-latitudes, the MarshallPalmer (1948) distribution is a reasonable approximation (except a very
small drop sizes – Fig.17):
N ( D )  N 0 e  D
where the slope factor depends on rainfall rate :
 ( R)  41R 0.21
Intercept is given by :
N 0  0.08 cm  4
size distributions
Where N(D)dD is no. ofMeasured
drops sizedrop
D to D+dD
per unit vol; R is in mm
-1
and M-P exponential curves
hr
4
10
Break-up may account for the negative-exponential form, raindrops
NDlimited in size to D ~ 3mm.
being
-3
-1
m mm
102
1
5m
m
m
25
m
m
hr -1
m
hr -1
-1
hr
100
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0
Fig.17
1
2
D mm
3
4
5
Cloud physics
8. Cloud seeding
The introduction to this Note suggested that an early impetus for cloud
physics studies was the desire to unravel the possible mechanisms
behind seeding clouds with various materials to encourage
precipitation. Scientist are much more cautious about their claims now
(‘would the cloud have precipitated anyway?’), although there are well
documented experiments claiming to confirm that, under certain
circumstances, seeding has been effective.
Two seeding agents are: silver iodide (AgI), which heterogeneously
nucleates ice at a temperature as high as –4oC, and ‘dry ice’ (solid CO2),
which has an equilibrium sublimation temperture of -78 oC.
The basic premis it that the precipitation-forming processes already
present (collision/coalescence, ice processes from natural nuclei) are
inefficient. Both seeding agents will vastly increase the numbers of ice
crystals in the cloud, which will grow at the expense of the water
droplets by the Bergeron-Findeisen mechanism (Section 6, Fig.11). The
number of ice particles nucleated will be substantially less than the
number of water droplets initially in the cloud, so each ice particle can
grow to a relatively large size. The ice particles, being larger than the
liquid droplets, fall faster than the droplets they replaced, and can fall to
the melting layer to form rain.
9. Lightning – cloud charging mechanisms
Mechanisms: Inductive – ‘charge stripping’ is important. Disintegration
of large water drops or ice crystals also occurs (Fig. 18).
Deformation & disintegration of raindrops
• For drops > 6mm diameter
aerodynamic pressures exceed
surface tension forces. Small scale
turbulence encourages instability
and breakup.
• A large falling drop flattens and
develops a depression in its base
before breaking up.
2cm
• Drop blows up to form expanding
‘bag’ supported by toroidal liquid
ring that later breaks into drops
• Inductive charge generation is
likely in this process.
Fig.18
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Thermo-electric effect
Supercooled water drop freezes on impact with
hailstone. Charge generated across ice-shell
+ ve carried
upwards on
splinters
• H+ (OH)-
+ C
ice
H+,(OH)-
Hailstone
-ve charge communicated
to hailstone
W -
- 20oC
---------
0oC
H+ ,(OH)-
More dissociation at W.
Diffusion of mobile H+
down concentration
gradient: - cold end + ve
----
Ice shell
Drop
trajectory
Supercooled drop
Fig.19
Non-inductive – a large number of mechanisms possible, particularly
those involving collisions between graupel and ice particles (Fig 19).
9.1 Cloud charge distribution
The classic ‘tripole’ charge distribution is apparent in the updraught
region (Fig.20). There is basically an upper positive pocket, with the
main negative beneath (at about –20 to –25oC), with a positive pocket
beneath (at ~ 0oC). A similar but more complex distribution is associated
Charge distribution in a thundercloud
Fig. 18
-------------------------+++++++
++++++++++++++++
++++++++++++++++
++++
-------- ---------
---------- ------------+++++
+++++
+++++
+++++
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with the downdraught.
Fig. 20
9.2 Lightning and cloud microphysics – a summary

Lightning and microphysics are closely linked

More observational data are required

Models must include greater complexity:
-
more charging mechanisms
-
improved interactions between ‘processes’
-
couplings developed with other models.
10. References and further reading
Berry, E et al.
1974 An analysis of cloud droplet growth by
collection. J Atmos.Sci, 31, 1814-24 & 1825-31
Blyth, A & Latham J 1998 Comments on glaciation papers by Hobbs et al
Q.J.R.Met.S. 124, 1007-8
(*See
reply by Hobbs & Rangno)
Choularton, T et al
1998 A study of the effects of cloud processing of
aerosol on the microphysics of clouds.
Q.J.R.Met.S. 124, 1377-1389
Fletcher, N
1962 The physics of rainclouds,
Cambridge University Press.
Hobbs, P
1993 Aerosol-cloud-climate interactions
Academic Press (Ed. P V Hobbs)
Hobbs, P et al
1985
Particles in the lower troposphere over the
high plains of the United States.
J. Clim. Appl. Meteor. 24, 1344 - 1356.
Hobbs, P & Rangno* 1998 Reply to: ‘Comments on glaciation papers by
Hobbs et al’
Q.J.R.Met.S. 124, 1009-10
IPCC
1995, 2000 and sequels: The science of climate change
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Jonas, P & Mason.B
1982 Entrainment and the droplet spectrum in
cumulus clouds. Q.J.R.Met.S. 108, 857-869
Junge, C & McLaren 1971 Relationship of cloud nuclei spectra to aerosol
size distribution and composition.
J.Atmos. Sci., 28, 382-390
Ludlam, F
1980
Clouds and storms, Pub: Penn State
Marshall, J & Palmer,W1948 The distribution of raindrops with size
J. Meteor. 5, 165-166
Mason, B
1971 (2nd edition) The physics of clouds.
Oxford University Press
Mason, B
1975 (2nd edition) Clouds, rain and rainmaking
Cambridge University Press.
Mason, B
1996 The rapid glaciation of slightly supercooled
cumulus clouds. Q.J.R.Met.S. 122, 357-365
Mordy, W
1959 Computations of the growth by condensation of
a population of cloud droplets. Tellus, 11, 16-44
Mossop, S
1985 There are two papers on ice particle
multiplication in: Q.J.R.Met.S. 111, 113-124
and 183-198
Osborne, S
1996 The processing of aerosols by warm
stratocumulus clouds.
MRF Internal Note No. 64
Pruppacher, H & Jänicke 1995
Processing of water vapour and
aerosols by atmospheric clouds, a global estimate
Atmos. Res. 38, 283-295
Püschel, R
1995 Atmospheric aerosols
Composition, chemistry and climate of the
atmosphere. Int Thomson Publ Inc, 120-175
Rangno, A & Hobbs, P
1994 Ice particle concentrations and
precipitation development in small polar continental
cumuliform clouds.
Q.J.R.Met.S. 120, 573-601
Rogers, R and Yau, M
Physics,
Slingo, A et al
1989 (3rd edition) A short Course in Cloud
Pergamon Press.
1982 Aircraft observations of marine Sc during
JASIN. Q.J.R.Met.S. 108, 833-856
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Starr, J
1967 Deposition of particulate matter by hydrometeors
Q.J.R.Met.S. 93, (398),516-521
Starr, J
1967 Inertial impaction of particulates on bodies of
simple geometry. Ann.Occup.Hyg, 10, 349-361
Taylor,J & McHaffie,A
1994
Measurements of cloud susceptibility
J Atmos.Sci, 51, 1298-1306
Telford, J & Chai
1980 A new aspect of condensation theory.
Pure & Appl. Geophys. 118, 720 - 742.
Toon, O
1995 Modelling relationships between aerosol
properties and the direct/indirect effects of aerosols on
climate: Aerosol forcing of climate. John Wiley &
Sons.
Wang, P et al
1978 Effect of electric charges on the scavenging of
aerosols by clouds and small droplets
J. Atmos. Sci., 35(9), 1735-1743
Wilson, D & Ballard 1999 A microphysically-based precipitation scheme
for the UKMO Unified Model,
Q. J. R. Met. S. 125, 1607-1636
Woods, J
1965 Wake capture of water drops in air.
Q.J.R.Met.S. 91, 585-7
Of general interest:
Physical Characteristics of Water:
Met. O College Note, Mar-2000 (J Starr)
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