METR215: Advanced Physical Meteorology: Water Droplet Growth Condensation & Collision

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:


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)


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)


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


Growth slows down with increasing droplet size:


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.


Let’s now look at evolution of droplet size w/height in cloud

• supersaturation vs. height w/ pseudoadiabatic ascent:


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.


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



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.



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).


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.




Pseudoadiabatic Calculation (H.W.)

rv

LWC

Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992)


Data from U. Washington C-131

aircraft



aircraft




aircraft


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.


r1.0

1 s

Dry particle - CCN

wet haze droplet

activated CCN

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 .


log(NCCN )

k ~ 0.5 (clean air)

log s (%)

k ~ 0.8 (polluted air)


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:



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


If a Junge number distribution (e.g., w/=-3) held for CCN, such k’s not found experimentally



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 =>


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!


Reflectance LWC2

3 N1

3 z


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)


• 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.


• 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


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



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:



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 :



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



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


Terminal Velocity of Drops/Droplets:


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:



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.


R&Y,p. 132-133


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'




• 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


W/out condensation With condensation

R&Y, p. 144

nuclei are activatedcondensation growth

collision/coalescence growth

newly activated droplets (transient)


Water Droplet Growth - Collisional + Condensation, cont.


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

Documents

METR215: Advanced Physical Meteorology: Water Droplet Growth Condensation & Collision