Example Observations 21 November 1354 EST T 11

Example Observations 21 November 1354 EST T = 11. 7°C Scattered altocumulus (Ac) at ~6 km 22 Nov. 0800 EST T = 8. 9°C Temperature stable for hours. Mist, fog, overcast. Little diel cycle because of cloud cover and condensation. 22 Nov. 1400 EST T = 17. 8°C Broken cirrus at ~7 km. Radiative heating. Copyright © 2010 R. R. Dickerson & Z. Q. Li 1

AOSC 620 Lecture 23 Cloud Droplet Population Growth – Warm Cloud Precipitation (Rogers and Yau Chapt. 7) R. Dickerson, 2010 In the last class (Lecture 22) we showed how condensation leads to cloud droplets. How do droplets grow and become rain drops? Balance among: 1. Molecular diffusion of H 2 O 2. Heat conduction 3. Curvature (surface tension) and solute effects (Raoult’s Law). 4. Temp. dependence of vapor pressure in CC Equation. Copyright © 2010 R. R. Dickerson & Z. Q. Li 2

Final Set of Growth Equations • Mass diffusion to the droplet • Conduction of latent heat away • Combined curvature and solute effects • Clausius-Clapeyron equation

Molecular diffusion vs. eddy diffusion. D ~ 2 x 10 -5 m 2 s-1 t = Xd 2/D Kz ~ 20 m 2 s-1 t = XK 2/Kz How much travel in 1. 00 min? 60 = Xd 2/D → XD = (120 x 10 -5)½ = 3. 5 mm → XK = (60 x 20)½ = 350 m • Eddy diffusion is much faster than molecular diffusion, but eddies cannot mix water vapor toward small droplets. Droplets move together.

How do we make it rain? If all growth were due to condensation there would be never be rain. We first considered stationary droplets, now a parcel moving along a pseudoadiabat. They we discuss how droplets collide sometimes. Coalescence: Consider warm clouds, i. e. , T ≥ 0ºC, no ice. Condensation can get droplets to 20 mm. collection condensation ~20 Radius ↑ (mm) Time →

Assumptions • Isolated, spherical water droplet of mass M, radius r and density w • Droplet is growing by the diffusion of water vapor to the surface • The temperature T and water vapor density v of the remote environment remain constant • A steady state diffusion field is established around the droplet so that the mass of water vapor diffusing across any spherical surface of radius R centered on the droplet is independent of R and time t.

Growth of a Population of Cloud Droplets Interaction between droplets? r ≈ 7 µm; x ≈ 1000 µm; vt ≈ 0. 5 cm s-1 • The separation between droplets is so great, relative to their sizes, that any interaction between droplets is negligible (not counting collisions). • Each growing droplet experiences the same environment. That is, droplets influence each other only in their combined effect on the common environment.

Qualitative Description of Condensation in a Rising Current of Air • As an initially unsaturated air parcel rises and expands approximately adiabatically, the saturation ratio S = (e / es ) increases and nuclei swell. • After the saturation level (LCL) is reached, condensation begins to occur on the largest, most active nuclei.

Qualitative Description of Condensation in a Rising Current of Air - cont. • S continues to increase and more nuclei are activated and begin to grow as droplets. However, the rate of increase in S is slower than above because the growing droplets are rapidly removing the excess water vapor from the parcel. • Since the large droplets remove the water vapor more quickly than the smallest ones, the excess vapor is soon being removed from the air as fast as it is supplied from expansion. Then S decreases toward unity.

Growth Equations for a Population of Nuclei In order to calculate the rate of change of S with time, one must consider: • the updraft speed • the condensation nuclei population • the growth equations for each nuclei. We’ll assume 1. a parcel rising with constant vertical speed dz/dt = U; 2. no mixing with the environment; 3. no collisions between droplets.

Growth Equations for a Population of Nuclei I

Growth Equations - cont. II

Growth Equations - cont. III i. e. , if there is condensation, w changes, and T changes And the parcel is more nearly saturated adiabatically.

Growth Equations - cont. Inserting III into II and then II into I

Growth Equations - cont.

Growth Equations - cont.

Growth Equations – final The change in Saturation ratio, S, is Production minus Condensation (loss). R&Y Eq. 7. 23 Eq. 7. 24

Equation for Liquid Water Content where ni is the number of condensation nuclei of size i per unit volume, and ri is the radius of a droplet formed on the ith nucleus.

Liquid Water Content - cont.

Growth Equations

Q 1 is increase in S due to adiabatic cooling; Q 2 is the decrease due to condensation.

Growth Calculations • Thus, the analysis yields a set of simultaneous equations. • Given initial conditions of T, p, w, U and ni, we would like to solve for S as a function of time (height in a cloud) ri as a function of time/height

Results of Calculation of droplet spectrum Initial Conditions U - 15 cm s-1 moderate concentration of nuclei at cloud base Results Fig. 7. 3 from Rogers (1989) • All droplets begin to grow as they ascend • S* reaches a max ~ 10 m above cloud base • Size distribution is very narrow • Largest droplets descend

Results of Calculations - cont. from Rogers (1989 Initial conditions Population of Na. Cl nuclei with Nc = 650 • S*0. 7 cm-3 U = 2 m s-1 (solid) and 0. 5 m s-1 (dashed)

The variable im/Ms is just the molar mixing ratios. The droplets formed from small CCN fail to activate; they evaporate back into haze. The From Wallace & Hobbs

Overall Results (what does our cloud look like? ) • Sharp rise and gentle settling of supersaturation. • S* reaches its peak within 100 m of cloud base. Thus the cloud droplet concentration is determined at low levels in the cloud. • The maximum of S* increases with U as does the number of nuclei activated. • Rapid increase of droplet concentration to a steady value reached at point of maximum supersaturation • Narrow droplet size distribution.

Vertical variations of cloud droplet sizes and liquid water density for low-level stratiform clouds compiled from various in-situ measurements. Note the general linear increasing trends! After Miles et al. (JAS, 2000 JAN)

Vertical variations of cloud number concentration. Note the difference between continental and marine clouds. Marine clouds has much smaller # of droplets which does not change much with height. The opposite is the case for continental clouds. ? After Miles et al. (JAS, 2000 JAN)

Problems with Diffusion Growth Theory • Kinetic effects - The heat mass and momentum transfer equations used are valid only for drops much larger than the molecular mean free path of water vapor in air (~0. 06 mm). In this case, the field of water vapor and temperature may be regarded as continua so that the Maxwell continuum approximation can be used to determine the transfer of water and heat. • When kinetic effects are included, the size reached by a drop is smaller at a given time than our model. The kinetic corrections result in a relatively broader spectrum of sizes.

Ventilation effects - When a drop is large enough to fall through the air with a significant speed, the vapor field becomes distorted from the spherical symmetry used in our model. These effects are negligibly small for cloud droplets, although they can become more significant for precipitation.

Problems - cont. • Nonstationary growth - The vapor and temperature fields are not steady because the surface of the droplet is expanding or contracting. However, calculations show that molecular diffusion quickly establishes a field around the droplet that corresponds to the steady state solution. • Unsteady updraft - Calculations show the results are relatively insensitive to the updraft structure because there is a strong correlation between Smax and U. • Statistical effects - Mixing and sedimentation prevent a population of droplets from staying together indefinitely (Rogers, 1989). Some drops may grow faster than others because they experience higher than average S’s. Not yet accepted conclusively.