Date post: | 18-Jan-2016 |
Category: |
Documents |
Upload: | hortense-lynch |
View: | 221 times |
Download: | 0 times |
METR215: Advanced Physical Meteorology: Water Droplet Growth Condensation & Collision
• Condensational growth: diffusion of vapor to droplet
• Collisional growth: collision and coalescence (accretion, coagulation) between droplets
Water Droplet Growth - Condensation
Flux of vapor to droplet (schematic shows “net flux” of vapor towards droplet, i.e., droplet grows)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Need to consider:
1. Vapor flux due to gradient between saturation vapor pressure at droplet surface and environment (at ∞).
2. Effect of Latent heat effecting droplet saturation vapor pressure (equilibrium temperature accounting for heat flux away from droplet).
rdr
dt G(T) senv
a(T)
r
b
r3
rdr
dt G(T) senv
For large droplets:
Solution to diffusional drop growth equation:
Water Droplet Growth - Condensation
Integrate w.r.t. t (r0=radius at t=0 when particle nucleates):
r(t) ro2 2G(T) senv t
(similar to R&Y Eq. 7.18)
Water Droplet Growth - Condensation
Evolution of droplet size spectra w/time (w/T∞ dependence for G understood):
large droplets : r(t) ro2 2G senv t
T (C) G (cm2/s)* G (µm2/s)
-10 3.5 x 10-9 0.35
0 6.0 x 10-9 0.60
10 9.0 x 10-9 0.90
20 12.3 x 10-9 12.3
With senv in % (note this is the value after nucleation, << smax):
T=10C, s=0.05% => for small r0:
r ~ 18 µm after 1 hour (3600 s)r ~ 62 µm after 12 hours
* From Twomey, p. 103.
Diffusional growth can’t explain production of precipitation sizes!
G can be considered as constant with TSee R&Y Fig.7.1
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
What cloud drop size drop constitutes rain?
• For s < 0, dr/dt < 0. How far does drop fall before it evaporates?
large drops fall much further than small drops before evaporating.
VT ~ r2
VT t r2 r2 r4
(“Stokes” regime, Re <1, ~ 1 cm s-1 for 10 µm drop)
Water Droplet Growth - Condensation
rdr
dt G(T) senv constant t r2
• Approx. falling distance before evaporating:
r (mm) VT(m-s-1) 1km/VT (min)
0.01 0.01 ---
0.1 0.3 56
1 4.0 4.2
3 8.1 2.1
Minimum time since r evaporating as it falls
Water Droplet Growth - Condensation
Growth slows down with increasing droplet size:
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
R&Y, p. 111
large droplets : dr
dt~
G senv r
Since large droplets grow slower, there is a narrowing of the size distribution with time.
Water Droplet Growth - Condensation
Let’s now look at evolution of droplet size w/height in cloud
• supersaturation vs. height w/ pseudoadiabatic ascent:
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
ds
dt cooling from expansion - loss due to condensation + ...
Example calc., R&Y, p. 106, w= 15 cm/s:
• s reaches a maximum (smax), typically
just above cloud-base.
• Smallest drops grow slightly, but can then evaporate after smax reached.
• Larger drops are activated; grow rapidly in region of high S; drop spectrum narrows due to parabolic form of growth equation.
solute mass
s(z)
Example calc., R&Y, p. 109, w= 0.5, 2.0 m/s:
• since s - 1 controls the number of activated condensation nuclei, this number is determined in the lowest cloud layer.
• drops compete for moisture aloft; simple modeling shows a limiting supersaturation of ~ 0.5%.
Evolution of droplet size w/height in cloud, cont.
Water Droplet Growth - Condensation
Corrections to previous development:
Ventilation Effects
• increases overall rate of heat & vapor transfer
Ventilation coefficient, f :
f = 1.06 for r = 20 µm; effect not significant except for rain
f 1 0.09Re 0 Re 2.5
0.78 0.28 Re0.5 Re 2.5
Re rv
; µ dynamic viscosity, v velocity
Water Droplet Growth - Condensation
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Corrections to previous development:
Kinetic Effects
Continuum theory, where r >> mean free path of air molecules (~0.06 µm at sea level). Molecular collision theory, where r << mean free path of molecules
• newly-formed drops (0.1 to 1 µm) fall between these regimes. • kinetic effects tend to retard growth of smallest drops, leading to broader spectrum.
Water Droplet Growth - Condensation
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Diffusional growth summary (!!):
• Accounted for vapor and thermal fluxes to/away from droplet.
• Growth slows down as droplets get larger, size distribution narrows.
• Initial nucleated droplet size distribution depends on CCN spectrum & ds/dt seen by air parcel.
• Inefficient mechanism for generating large precipitation sized cloud drops (requires hours). Condensation does not account for precipitation (collision/coalescence is the needed for “warm” clouds - to be discussed).
Water Droplet Growth - Condensation
Many shallow clouds with small updrafts (e.g., Sc), never achieve precipitation sized drops. Without the onset of collision/coalescence, the droplet concentration in these clouds (N) is often governed by the initial nucleation concentration. Let’s look at examples, starting with previous pseudoadiabatic calculations.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Condensation
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Pseudoadiabatic Calculation (H.W.)
rv
LWC
Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Data from U. Washington C-131
aircraft
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Data from U. Washington C-131
aircraft
Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Data from U. Washington C-131
aircraft
Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992)
How can we approximate N for such clouds, and what does this tell us about the effect of aerosol (CCN) on cloud microphysics?
Approximation (analytic) for smax, N in developing cloud, no entrainment (from Twomey):
1. Need relationship between N and s => CCN(s) relationship is needed (i.e., equation for concentration of total nucleated haze particles vs. s, referred to as the CCN spectrum).
2. Determine smax.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
r1.0
1 s
Dry particle - CCN
wet haze droplet
activated CCN
Water Droplet Growth - microphysics approx.
Water Droplet Growth - microphysics approx.
CCN spectrum:
Measurements show that:
NCCN (s) c sk ,
where c = CCN concentration at s=1%.
If smax can be approximated for a rising air parcel, then the number of cloud droplets is:
N c smaxk .
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
log(NCCN )
k ~ 0.5 (clean air)
log s (%)
k ~ 0.8 (polluted air)
Water Droplet Growth - microphysics approx.
ds
dt1 s
A (cooling from dryadiabatic expansion)
B (vol. change decreases env [w≠w(z) => ws incr. with z])–
C (vapor depletion due to droplet growth)
D (latent heat warms droplet, air & es increases)+– [ ]
– [ ]
dz
dt
d LWC
dt
Note: pseudoadiabatic lapse rate keeps RH=100%, s=0, ds/dt=0. No entrainment of dry air (mixing), no turbulent mixing, etc.
Twomey showed (1959) that an upper bound on smax is:
smax (A B)dz
dt
3
2(k2) ck
1
k2
Approx. for smax:
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - microphysics approx.
Therefore, the upper bound on is determined from is:
N c2
k2 dz
dt
3k
2(k2)
N c smaxk
• k = 1
• k = 1/2
• k ≥ 2
N c 0.8 dz
dt0.3 , proportional to c
N dz
dt0.75 , proportional to updraft velocity
N c 0.67 dz
dt0.5 , depends on c and updraft velocity
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
If a Junge number distribution (e.g., w/=-3) held for CCN, such k’s not found experimentally
Water Droplet Growth - microphysics approx.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Very important result!
1. NCCN controls cloud microphysics for clouds with relatively small updraft velocities (e.g., stratiform clouds).
2. Increase NCCN (e.g., by pollution), then N will also increase (by about the same fractional amount if pollution doesn’t modify k).
t
s(t)
clean air (e.g., maritime)
“dirty” air (e.g., continental)
smaxclean
smaxdirty
Note:
smaxclean smax
dirty
cclean cdirt =>
Water Droplet Growth - microphysics approx.
Ship Tracks - example of increase in CCN modifying cloud microphysics
• Cloud reflectance proportional to total cloud droplet cross-sectional area per unit area (in VIS/NIR part of solar spectrum) or the cloud optical thickness:
So what happens when CCN increase?
Reflectance r2 N z
• Constraint: Assume LWC(z) of cloud remains the same as CCN increases (i.e., no coalescence/precipitation). Then an increase in N implies droplet sizes must be reduced => larger droplet cross-sectional area and R increases. Cloud is more reflective in satellite imagery!
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Reflectance LWC2
3 N1
3 z
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Cloud-aerosol interactions ex.: ship tracks (27 Jan. 2003, N. Atlantic)
MODIS (MODerate resolution Imaging Spectroradiometer)
Pseudoadiabatic Calculations(Parcel model of Feingold & Heymsfield, JAS, 49, 1992)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
• Droplets collide and coalesce (accrete, merge, coagulate) with other droplets.
• Collisions governed primarily by different fall velocities between small and large droplets (ignoring turbulence and other non-gravitational forcing).
• Collisions enhanced as droplets grow and differential fall velocities increase.
• Not necessarily a very efficient process (requires relatively long times for large precipitation size drops to form).
• Rain drops are those large enough to fall out and survive trip to the ground without evaporating in lower/dryer layers of the atmosphere.
Water Droplet Growth - Collisions
concept
Homogeneous Mixing: time scale of drop evaporation/equilibrium much longer relative to mixing process. All drops quickly exposed to “entrained” dry air, and evaporate and reach a new equilibrium together. Dilution broadens small droplet spectrum, but can’t create large droplets.
Inhomogeneous Mixing: time scale of drop evaporation/equilibrium much shorter than relative to turbulent mixing process. Small sub-volumes of cloud air have different levels of dilution. Reduction of droplet sizes in some sub-volumes, little change in others.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
• Droplets collide and coalesce (accrete, merge, coagulate) with other droplets. Collisions require different fall velocities between small and large droplets (ignoring turbulence and other non-gravitational forcing).
• Diffusional growth gives narrow size distribution. Turns out that it’s a highly non-linear process, only need only need 1 in 105 drops with r ~ 20 µm to get process rolling.
• How to get size differences? One possibility - mixing.
Water Droplet Growth - Collisions
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Approach:
• We begin with a continuum approach (small droplets are uniformly distributed, such that any volume of air - no matter how small - has a proportional amount of liquid water.
• A full stochastic equation is necessary for proper modeling (accounts for probabilities associated with the “fortunate few” large drops that dominate growth).
• Neither approach accounts for cloud inhomogeneities (regions of larger LWC) that appear important in “warm cloud” rain formation.
Water Droplet Growth - Collisional Growth
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
VT(R)
R
VT(r)
"capture"distance VT R VT r t
d(sweepout volume)
dt R r 2 VT R VT r R2 VT R
collected mass : dm
dt R2 VT R LWC
also : dm
dt
d
dt
4
3r3
l 4r2 dr
dtl
substitution :dR
dt
VT R LWC
4l
(increases w/R,
vs. condensation wheredR/dt ~ 1/R)
Continuum collection:
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
dR
dt
3l
R r
R
2
VT R VT r r3 n(r) dr
Integrating over size distribution of small droplets, r, and keeping R+r terms :
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Accounting for collection efficiency, E(R,r):
If small droplet too small or too far center of collector drop, then capture won’t occur.
• E is small for very small r/R, independent of R.
• E increases with r/R up to r/R ~ 0.6
• For r/R > 0.6, difference is drop terminal velocities is very small.
–drop interaction takes a long time, flow fields interact strongly and droplet can be deflected.–droplet falling behind collector drop can get drawn into the wake of the collector; “wake capture” can lead to E > 1 for r/R ≈ 1.
dR
dt
3l
R r
R
2
VT R VT r E(R,r) r3 n(r) dr
PHYS 622 - Clouds, spring ‘04, lect.4, PlatnickPHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Collection Efficiency, E(R,r):
R&Y, p. 130
• differences in fall speed lead to conditions for capture.
• terminal velocity condition:
constant fall velocity VT
where r is the drop radius L is the density of liquid water g is the acceleration of gravity
is the dynamic viscosity of fluid is the Reynolds’ number. u is the drop velocity (relative to air) CD is the drag coefficient
FG FD
FG 4
3 r3 L g
FD 6 ruCD Re
24
Re 2u r
VT(R)VT(r)
FD
FG
Water Droplet Growth - Collisional Growth
Terminal Velocity of Drops/Droplets:
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Low Re; Stokes’ Law: r < 30 m
High Re: 0.6 mm < r < 2 mm
Intermediate Re: 40 m < r < 0.6 mm
CD ~ const. VT = k2 r 0.5 ; k2 = 2x 103 o
0.5
cm0.5 s-1
o = 1.2 kg m-3
CD Re
24 1 VT 2r 2 gL
9 k1 r2 ;
k1 = 1.19 x 106 cm-1 s-1
VT = k3 r ; k3 = 8 x 103 s-1
Terminal Velocity Regimes:
Water Droplet Growth - Collisional Growth
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
air parcel droplets
collector drop
• Fig. 8.4: collision/coalescence process starts out slowly, but VT and E increase rapidly with drop size, and soon collision/coalescence outpaces condensation growth.
• Fig. 8.6:– with increasing updraft speed, collector ascends to higher altitudes, and emerges as a larger raindrop.
– see at higher altitudes, smaller drops; lower altitudes, larger drops.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
R&Y,p. 132-133
Water Droplet Growth - Collisional Growth
Stochastic collection: account for distribution n(r) or n(m)
Collection Kernel: effective vol. swept out per unit time, for collisions between drops of mass and :
Probability that a drop of mass will collect a drop of mass in time dt:
K(m,m') R r 2 E(R,r) VT R VT r
m
m'
m
m'
P(m,m') K(m,m')n(m')dm' dt
dn(m)
dt n(m) K(m,m')n(m')dm'
1
2K(m',m m')n(m')n(m m')dm'
0
m
0
loss of -sized drops due to collection with other sized
drops
m formation of -sized drops from coalescence with
and drops (counting twice in integral -> factor of 1/2
m'
m
m m'
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
• Larger drops in initial spectrum become “collectors”, grow quickly and spawn second spectrum.
• Second spectrum grows at the expense of the first, and ,mode r increases with time.
Stochastic collection, example:
R&Y, p. 130,also see Fig. 8.11
Water Droplet Growth - Collisional + Condensation
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
W/out condensation With condensation
R&Y, p. 144
nuclei are activatedcondensation growth
collision/coalescence growth
newly activated droplets (transient)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional + Condensation, cont.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Cloud Inhomogeneity
Evolution of drop growth by coalescence very sensitive to LWC, due to non-linearity of stochastic equation. Non-uniformity in LWC can aid in production of rain-sized drops.
• Example (S. Twomey, JAS, 33, 720-723, 1976): see Fig. 2