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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.

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

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  1. 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

  2. Water Droplet Growth - Condensation Flux of vapor to droplet (schematic shows “net flux” of vapor towards droplet, i.e., droplet grows) Need to consider: Vapor flux due to gradient between saturation vapor pressure at droplet surface and environment (at ∞). Effect of Latent heat effecting droplet saturation vapor pressure (equilibrium temperature accounting for heat flux away from droplet). PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  3. Water Droplet Growth - Condensation Solution to diffusional drop growth equation: (similar to R&Y Eq. 7.18) For large droplets: Integrate w.r.t. t (r0=radius at t=0 when particle nucleates):

  4. Diffusional growth can’t explain production of precipitation sizes! Water Droplet Growth - Condensation Evolution of droplet size spectra w/time (w/T∞ dependence for G understood): With senv in % (note this is the value after nucleation, << smax): G can be considered as constant with T See R&Y Fig.7.1 *From Twomey, p. 103. T=10C, s=0.05% => for small r0: r ~ 18 µm after 1 hour (3600 s) r ~ 62 µm after 12 hours

  5. Water Droplet Growth - Condensation What cloud drop size drop constitutes rain? • For s < 0, dr/dt < 0. How far does drop fall before it evaporates? (“Stokes” regime, Re <1, ~ 1 cm s-1 for 10 µm drop) • Approx. falling distance before evaporating: • large drops fall much further than small drops before evaporating. Minimum time since r evaporating as it falls PHYS 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  6. Water Droplet Growth - Condensation Growth slows down with increasing droplet size: Since large droplets grow slower, there is a narrowing of the size distribution with time. R&Y, p. 111 PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  7. Water Droplet Growth - Condensation • Let’s now look at evolution of droplet size w/height in cloud • supersaturation vs. height w/ pseudoadiabatic ascent: 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. s(z) solute mass PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  8. Water Droplet Growth - Condensation Evolution of droplet size w/height in cloud, cont. 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%.

  9. µ dynamic viscosity, v velocity f= 1.06 for r = 20 µm; effect not significant except for rain Water Droplet Growth - Condensation Corrections to previous development: Ventilation Effects • increases overall rate of heat & vapor transfer Ventilation coefficient, f :

  10. Water Droplet Growth - Condensation 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. PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  11. Water Droplet Growth - Condensation • 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). PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  12. 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

  13. Pseudoadiabatic Calculation (H.W.) LWC rv PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  14. Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992) Data from U. Washington C-131 aircraft PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  15. Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992) Data from U. Washington C-131 aircraft PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  16. Example Microphysical Measurements in Marine Sc Clouds (ASTEX field campaign, near Azores, 1992) Data from U. Washington C-131 aircraft PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  17. activated CCN wet haze droplet 1.0 r Dry particle - CCN Water Droplet Growth - microphysics approx. 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): 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). Determine smax. PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  18. k ~ 0.8 (polluted air) k ~ 0.5 (clean air) Water Droplet Growth - microphysics approx. CCN spectrum: Measurements show that: where c = CCN concentration at s=1%. If smax can be approximated for a rising air parcel, then the number of cloud droplets is: PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  19. Water Droplet Growth - microphysics approx. Approx. for smax: B (vol. change decreases renv [w≠w(z) => ws incr. with z]) A (cooling from dry adiabatic expansion) ] – [ – D (latent heat warms droplet, air & es increases) C (vapor depletion due to droplet growth) ] – [ + 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: PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  20. Water Droplet Growth - microphysics approx. Therefore, the upper bound on is determined from is: • k = 1 • k = 1/2 • k ≥ 2 If a Junge number distribution (e.g., w/a=-3) held for CCN, such k’s not found experimentally PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  21. Water Droplet Growth - microphysics approx. Very important result! NCCN controls cloud microphysics for clouds with relatively small updraft velocities (e.g., stratiform clouds). Increase NCCN (e.g., by pollution), then N will also increase (by about the same fractional amount if pollution doesn’t modify k). clean air (e.g., maritime) “dirty” air (e.g., continental) t Note:  => PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  22. 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 622 - Clouds, spring ‘04, lect.4, Platnick

  23. Cloud-aerosol interactions ex.: ship tracks (27 Jan. 2003, N. Atlantic) MODIS (MODerate resolution Imaging Spectroradiometer) PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  24. Pseudoadiabatic Calculations (Parcel model of Feingold & Heymsfield, JAS, 49, 1992) PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  25. Water Droplet Growth - Collisions • 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. concept

  26. Water Droplet Growth - Collisions • 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. • 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

  27. Water Droplet Growth - Collisional Growth • 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. PHYS 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  28. R VT(R) VT(r) (increases w/R, vs. condensation where dR/dt ~ 1/R) Water Droplet Growth - Collisional Growth Continuum collection: PHYS 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  29. Water Droplet Growth - Collisional Growth Integrating over size distribution of small droplets, r, and keeping R+r terms : PHYS 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  30. 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 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  31. Water Droplet Growth - Collisional Growth Collection Efficiency, E(R,r): R&Y, p. 130 PHYS 622 - Clouds, spring ‘04, lect.4, Platnick PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  32. VT(R) VT(r) FD FG Water Droplet Growth - Collisional Growth Terminal Velocity of Drops/Droplets: • differences in fall speed lead to conditions for capture. • terminal velocity condition: constant fall velocity VT where r is the drop radius rL is the density of liquid water g is the acceleration of gravity m is the dynamic viscosity of fluid is the Reynolds’ number. u is the drop velocity (relative to air) CD is the drag coefficient PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  33. Water Droplet Growth - Collisional Growth Terminal Velocity Regimes: Low Re;Stokes’ Law: r < 30 mm High Re: 0.6 mm < r < 2 mm Intermediate Re: 40 mm < r < 0.6 mm k1 = 1.19 x 106 cm-1 s-1 PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  34. droplets air parcel R&Y, p. 132-133 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

  35. 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 Water Droplet Growth - Collisional Growth PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  36. Water Droplet Growth - Collisional Growth Stochastic collection, example: R&Y, p. 130, also see Fig. 8.11 • 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. PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  37. Water Droplet Growth - Collisional + Condensation W/out condensation With condensation R&Y, p. 144 PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  38. Water Droplet Growth - Collisional + Condensation, cont. newly activated droplets (transient) nuclei are activated collision/coalescence growth condensation growth PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

  39. 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 PHYS 622 - Clouds, spring ‘04, lect.4, Platnick

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