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 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).

For large droplets: Solution to diffusional drop growth equation: Water Droplet Growth - Condensation Integrate w.r.t. t (r 0 =radius at t=0 when particle nucleates): (similar to R&Y Eq. 7.18)

Water Droplet Growth - Condensation Evolution of droplet size spectra w/time (w/T ∞ dependence for G understood): T (C)G (cm 2 /s) * G (µm 2 /s) x x x x With s env in % (note this is the value after nucleation, << s max ): T=10C, s=0.05% => for small r 0 : r ~ 18 µm after 1 hour (3600 s) r ~ 62 µm after 12 hours * From Twomey, p Diffusional growth can’t explain production of precipitation sizes! G can be considered as constant with T See R&Y Fig.7.1

PHYS 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. (“Stokes” regime, Re <1, ~ 1 cm s -1 for 10 µm drop ) Water Droplet Growth - Condensation Approx. falling distance before evaporating: r (mm)V T (m-s -1 )1km/V T (min) Minimum time since r evaporating as it falls

Water Droplet Growth - Condensation Growth slows down with increasing droplet size: PHYS Clouds, spring ‘04, lect.4, Platnick R&Y, p. 111 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 Clouds, spring ‘04, lect.4, Platnick Example calc., R&Y, p. 106, w= 15 cm/s: s reaches a maximum (s max ), typically just above cloud-base. Smallest drops grow slightly, but can then evaporate after s max 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)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 µ dynamic viscosity, v velocity Water Droplet Growth - Condensation

PHYS 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 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 Clouds, spring ‘04, lect.4, Platnick Water Droplet Growth - Condensation

PHYS Clouds, spring ‘04, lect.4, Platnick Pseudoadiabatic Calculation (H.W.) rvrv LWC

Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992) PHYS Clouds, spring ‘04, lect.4, Platnick Data from U. Washington C-131 aircraft

PHYS 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 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 s max, 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 s max. PHYS Clouds, spring ‘04, lect.4, Platnick r 1.0 Dry particle - CCN wet haze droplet activated CCN Water Droplet Growth - microphysics approx.

CCN spectrum: Measurements show that: where c = CCN concentration at s=1%. If s max can be approximated for a rising air parcel, then the number of cloud droplets is: PHYS Clouds, spring ‘04, lect.4, Platnick k ~ 0.5 (clean air) k ~ 0.8 (polluted air)

Water Droplet Growth - microphysics approx. A (cooling from dry adiabatic expansion) B (vol. change decreases  env [w≠w(z) => w s incr. with z]) – C (vapor depletion due to droplet growth) D (latent heat warms droplet, air & e s increases) + – [ ] ] 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 s max is: Approx. for s max : PHYS Clouds, spring ‘04, lect.4, Platnick

Water Droplet Growth - microphysics approx. Therefore, the upper bound on is determined from is: k = 1 k = 1/2 k ≥ 2 PHYS 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 Clouds, spring ‘04, lect.4, Platnick Very important result! 1. N CCN controls cloud microphysics for clouds with relatively small updraft velocities (e.g., stratiform clouds). 2. Increase N CCN (e.g., by pollution), then N will also increase (by about the same fractional amount if pollution doesn’t modify k). t clean air (e.g., maritime) “dirty” air (e.g., continental) Note: =>

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? 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 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 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 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 10 5 drops with r ~ 20 µm to get process rolling. How to get size differences? One possibility - mixing. Water Droplet Growth - Collisions

PHYS 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 Clouds, spring ‘04, lect.4, Platnick Water Droplet Growth - Collisional Growth VT(R)VT(R) R VT(r)VT(r) (increases w/R, vs. condensation where dR/dt ~ 1/R) Continuum collection:

PHYS Clouds, spring ‘04, lect.4, Platnick Water Droplet Growth - Collisional Growth Integrating over size distribution of small droplets, r, and keeping R+r terms :

PHYS 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.

PHYS 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 V T 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) C D is the drag coefficient VT(R)VT(R) VT(r)VT(r) FDFD FGFG Water Droplet Growth - Collisional Growth Terminal Velocity of Drops/Droplets: PHYS 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 k 1 = 1.19 x 10 6 cm -1 s -1 Terminal Velocity Regimes: Water Droplet Growth - Collisional Growth PHYS Clouds, spring ‘04, lect.4, Platnick

air parcel droplets collector drop Fig. 8.4: collision/coalescence process starts out slowly, but V T 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 Clouds, spring ‘04, lect.4, Platnick R&Y, p

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: loss of -sized drops due to collection with other sized drops formation of -sized drops from coalescence with and drops (counting twice in integral -> factor of 1/2 PHYS Clouds, spring ‘04, lect.4, Platnick

Water Droplet Growth - Collisional Growth PHYS 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 Clouds, spring ‘04, lect.4, Platnick W/out condensationWith condensation R&Y, p. 144

nuclei are activated condensation growth collision/coalescence growth newly activated droplets (transient) PHYS Clouds, spring ‘04, lect.4, Platnick Water Droplet Growth - Collisional + Condensation, cont.

PHYS 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, , 1976): see Fig. 2