Twelve Lectures on Cloud Physics

Bjorn Stevens

Winter Semester 2010-2011

Contents

1 Lecture 1: Clouds–An Overview 31.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 What is a cloud? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Why are we interested in clouds? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Cloud classification schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Lecture 2: Thermodynamic Basics 62.1 Thermodynamics: A brief review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Intensive, Extensive, and specific variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Thermodynamic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 The many variables of atmospheric thermodynamics . . . . . . . . . . . . . . . . . 8

2.4 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Lecture 3: Droplet Activation 113.1 Supersaturation over curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Solute effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The Kohler equation and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Lecture 4: Further Properties of an Isolated Drop 164.1 Diffusional growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Temperature corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.2 Drop size effects on droplet growth . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Terminal fall speeds of drops and droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Lecture 5: Populations of Particles 225.1 Converting distribution and density functions . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Time derivatives of densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Common distributions and their density functions . . . . . . . . . . . . . . . . . . . . . . . 25

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6 Lecture 6: Collection through Collision and Coalescence 266.1 Gravitational collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Smoluchowski (stochastic) collection equation . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 The collection kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Lecture 7: Warm clouds and drop spectral evolution 317.1 Exogenous theories of warm rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.1.1 Giant CCN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.1.2 Drop spectral preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 Endogenous theories of warm rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2.1 Turbulence enhancement to collision efficiencies . . . . . . . . . . . . . . . . . . . 327.2.2 Turbulent mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8 Lecture 8: Atmospheric ice and its initiation 348.1 The molecular structure of water and ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.2 Ice initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8.2.1 Homogeneous nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2.2 Heterogeneous nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Lecture 9: Ice crystal growth 399.1 Diffusional growth theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Ice crystal terminal fall speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Growth by collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4 Further ice-microphysical processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10 Lecture 10: The atmospheric aerosol 4210.1 Types of aerosol particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.2 Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.3 Aerosol Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.4 Aerosol Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11 Lecture 11: Microphysical models of cloud and aerosol 4711.1 Particle based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.2 Distribution based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.3 Parametric distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12 Lecture 12: Aerosol and cloud measurement systems 4912.1 In situ Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.1.1 Single particle sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.1.2 Expansion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.1.3 Particle Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.1.4 Particle Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.2 Active remote sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2.1 Cloud and Precipitation Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2.2 Lidars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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1 Lecture 1: Clouds–An Overview

1.1 Organization

The important points to remember here are: the grade is based on the oral exam, the oral exam is based onthe exercises. So working through the exercises is critical. The other important points are covered on thesyllabus, or course Einleitung.

1.2 What is a cloud?

When most people think of a cloud they know what it is, a white puffy thing in the sky. But when pushed tobe precise it is more difficult. Formally speaking a cloud is an aerosol, that is a suspension of one phase ofmatter in another. The glossary of the American Meteorological Society defines an aerosol as follows:

A colloidal system in which the dispersed phase is composed of either solid or liquid parti-cles, and in which the dispersion medium is some gas, usually air.

A characteristic of an aerosol is that it is disperse. It is not one thing, but many things, or many repetitions ofthe same thing, dispersed in space. This makes the boundary of the aerosol, or a cloud, somewhat difficultto define objectively and precisely.

But clouds are a special type of aerosol, so special in fact that we rarely speak of a cloud as being anaerosol. Clouds are a type of aerosol that comes into being when the atmosphere becomes supersaturatedwith respect to water, and cloud particles grow rapidly and become visibly apparent in a way that the aerosolin a subsaturated environment rarely is. This difference encourages the tendency to associate the atmosphericaerosol with only the smallest particles, traditionally those under 1 µ m, and the growing particles that arefound in water saturated environments as clouds. This distinction encourages one to speak of clouds assingular, compact, entities in contrast to the disperse atmospheric aerosol. However a cloud’s origin as acomponent of the atmospheric aerosol lingers in attempts to define “a cloud” objectively.

10-9 10-6 10-3 10-1 Diameter [m]

Freshly NucleatedAerosol Particles

Cloud condensation nuclei

Cloud droplets Drizzle droplets Rain drops

Graupel HailIce Crystals Snow flakes

Sea-salt &Mineral dust

3.5 10-6 3.5 10-2 3.5 10 50 Fallspeed [ms-1]

long lived, abundant , suspended short lived, precipitation

Cloud PrecipitationAerosol

more complex

Figure 1: Types of particles in the atmosphere, organized by size. Note that many of the particles span arange of size, and only typical sizes are given to place one type of particle in relation to another. Particlesare also ordered with respect to complexity. Liquid water droplets and drops tend to have the same shapeand composition, the shape of ice varies making it more complex, for the aerosol but the composition andthe shape/mixture state can be variable.

The ability of the particles that constitute a cloud to grow in a supersaturated environment leads to asequence of events that can grow particles sufficiently large to efficiently scatter light, and eventually so

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large as to efficiently precipitate from the atmosphere. Cloud particles are hydrometeors, and a subset ofthese form precipitation–in German Niederschlag. The study of how cloud particles come into being, howtheir distribution effects the transfer of radiant energy, and how they transform themselves into precipitationis the subject of cloud physics. How hydrometeors fit into the broader class of particles that one finds in theatmosphere is illustrated in Fig. 1.

This figure emphasizes that the atmosphere suspends many forms of particulate matter. On scales ofa few nanometers, one finds freshly nucleated aerosol particles, while hail stones have been documentedto grow to sizes of tens of centimeters in diameter. Thus particles in the atmosphere span a range of sizesof as much as eight orders of magnitude and the mass of atmospheric particulate matter spans a range ofscales that is more than twenty orders of magnitude. Small particles sediment with a terminal velocity thatis proportional to their diameter squared, hence a factor of ten in diameter means a factor of one hundred inthe time it takes a particle to settle and fall out of the sky. Very small particles effectively never fall fromthe sky, and are only removed by collisions with larger particles, or because they grow by other means tosizes large enough to effectively fall from the sky. While large particles are rare, as once form they rapidlyprecipitate to the surface.

Clouds also get their meaning because we can see them, or we feel their presence through their emissionof radiation which keeps the ground from cooling on a cloudy night. Hence an important part of whatmakes a cloud are its radiative properties, its propensity to scatter visible radiation and absorb and emitinfrared radiation. The scattering of visible radiation depends both on the amount of suspended water mass,and the size of the suspended particles, while the efficacy of clouds in absorbing and emitting infraredradiation depends primarily on the suspended water mass. While the suspended water mass, sometimescalled the liquid water path, is a cloud macroscopic parameter, largely controlled by dynamical processes, thecharacteristic drop size is a microphysical parameter and can be strongly influenced by cloud microphysicalprocesses.

1.3 Why are we interested in clouds?

Outside of poetic motivations there are principally two reasons why we are interested in clouds:

1. Clouds couple to the water cycle because they are the vessels in which precipitation develops.

2. Clouds couple to the radiative balance because they interact strongly with both short and long-waveradiation.

The importance of the water cycle, and the role clouds play in it should be self evident. Although hereit is important to say that the role microphysical processes play in the water cycle is less clear. Cloudphysics owes much of its origins to attempts, dating to the middle part of the last century, to artificiallyinfluence precipitation formation and weather. The basic idea was that by altering cloud microphysicalprocesses it might be possible to make clouds rain more, or less effectively, thereby bringing needed rainfallto dry regions, or perhaps limiting the negative impacts of severe whether. However, the link between cloudmicrophysical processes and rainfall have been difficult to establish, in part because if the atmosphere isdetermined to precipitate it manages to find some microphysical pathway to do so. Hence the importanceof cloud microphysical relative to cloud macrophysical processes has proven difficult to establish in anygeneral sense.

Radiation, as mentioned earlier is also an important reason for studying clouds. Clouds reflect significantamounts of solar radiation. As much as 50 Wm−2 on an annual and global average. This is a large number,more than a factor of ten larger than the radiative forcing associated with a doubling of CO2 concentrations

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in the atmosphere. This tendency of clouds to reflect solar radiation warms the planet, and is called thealbedo effect. or the shortwave cloud radiative effect, or sometimes simply “shortwave cloud forcing.”

This strong tendency of clouds to cool the surface is partially compensated by their greenhouse effect.By absorbing thermal radiation emitted at high temperatures (characteristic of the surface) and re-emittingit at colder temperatures (characteristic of the clouds) the net amount of thermal radiation emitted to spaceis reduced, thus acting to reduce the planets ability to cool itself. This is a warming, or greenhouse, effect,but can also be called the longwave cloud radiative effect, or longwave cloud radiative forcing. Globally andannually averaged the effect is about 25 Wm−2, thereby offsetting by about half the effect of cooling due tothe cloud albedo effect.

Ironically modern interest in cloud physics is also influenced by a desire (not shared by this author) toartificially modify clouds so as to control the Earth’s radiation balance. Indeed a number of schemes havebeen proposed whereby global warming associated with increased concentrations of greenhouse gases mightbe offset through the deliberate modification of cloud optical properties. Whether this is feasible, let alonesensible remains an area of vigorous scientific debate.

1.4 Cloud classification schemes

Figure 2: A stratocumulus cloud photographed from above during the DYCOMS field study which tookplace over the northeast Pacific in July 2001. Here the corrugated patterning of the stratiform cloud sheetreflects the underlying convective motions that result from the radiative cooling of the cloud top. This photowas taken by Gabor Vali from the cockpit of the NCAR-NSF C130 research aircraft.

The earliest and best know cloud classification scheme is that proposed by Luke Howard in 1804.

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Howard’s scheme attempts a Linnaean type classification of clouds into groups or combinations of groups.The building blocks of Howards classification are the familiar cloud types of stratus, cumulus and cirrus.The enduring contribution of Howard’s classification comes from the fact is that these basic cloud types re-flect underlying processes that govern the cloud formation. The wispy character of cirrus clouds comes fromthe ice that underlies their formation, and its tendency to grow quickly and fall out of the cloud in streaks.Stratus reflect the cooling associated with the uplifting of whole layers of the atmosphere, while the puffycharacter of cumulus reflect their origins as local convective elements whose turbulent character gives thesecloud their cumuliform character. Naturally such processes do not exist in exclusion, hence mixtures of suchprocesses, for instance convective motions in layer clouds, gives clouds whose character is best describedthrough a combination of the Howard archetypes, in the present case stratocumulus (e.g., Fig. 2).

The emergence of satellite meteorology has lead to the development of new cloud classification schemes,based on what can be effectively be measured by satellite. One such popular scheme is that associated withthe International Satellite Cloud Climatology Project (ISCCP), wherein clouds are classified according totheir cloud-top temperature, which is well measured by infrared sensors, and their optical depth, which canbe readily measured by their visible albedo. This classification scheme maps well onto those processes thatmotivate our interest in clouds in the first place. Optically thick clouds with cold tops are readily associatedwith precipitation. High thin clouds have a very strong greenhouse effect, while low clouds (those with highcloud-top temperatures) which are optically thick are dominated by their albedo effect.

2 Lecture 2: Thermodynamic Basics

Elemental to the formation of clouds is the requirement of air to become saturated with respect to watervapor. There are two basic paths to saturation, either through moistening of air at a fixed temperature, asfor instance happens when cold air flows over the warm ocean, or through the cooling of air parcels, as forinstance accompanies the expansion of air through adiabatic uplift. Fundamental to descriptions of cloudprocesses are thus the basic concepts of atmospheric thermodynamics: roughly speaking the relationshipbetween pressure, temperature and the diabatic production of entropy, through external heating or mixing.

2.1 Thermodynamics: A brief review

To fix notation we express the first law of thermodynamics as a postulate which asserts the existence of aninternal energy. This internal energy, U along with the volume, V of the system and the mole number, Ni

of its chemical constituents completely characterizes the macroscopic character of the systems equilibriumstates.

A convenient form of the second law is to state that for any equilibrium system there exists an entropy, Sand that this entropy has the property that it is a function of the extensive parameters of a composite system,such that

S = S(U, V,N1, . . . , Nr),

where Ni denotes the mole number of the ith constituent. Further it is postulated that the entropy is additiveover constituent subsystems, that it is continuous and differentiable and a monotonically increasing functionof the energy, U, and that in the absence of internal constraints the value of the extensive parameters inequilibrium are those that maximize the entropy. These properties ensure that the entropy function can besolved for the internal energy, such that

U = U(S, V,N1, . . . , Nr).

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This expression is sometimes called the Fundamental Relation in energy form. In specific (per unit mole, orunit mass) form, it can be written as

u = u(s, v, n1, . . . , nr),

where ni = Ni/∑

j Nj is the mole fraction.It then follows that

dU = (∂U/∂S)V,Ni dS + (∂U/∂V )S,Ni dV +∑i

(∂U/∂Ni)S,V dNi.

The partial derivatives that occur in the above expression are called intensive parameters. They describeproperties of the system that are familiar with us, specifically the temperature, T ≡ (∂U/∂S)V,Ni , andthe pressure p ≡ −(∂U/∂V )S,Ni . Perhaps less familiar is the eletro-chemical potential µi ≡ (∂u/∂ni)s,v.Using these symbols to represent the partial derivatives yields a more recognizable expression for the differ-ential of the fundamental relation, namely

dU = TdS − pdV +∑i

µidNi. (1)

Because the internal energy is additive over its constituent systems

U(λS, λV, λNi) = λU.

By differentiating both sides with respect to λ, and recalling the definition of the partial derivatives, it isstraightforward to show that

U = TS − PV +∑i

µiNi.

This is called the Euler relation. By the laws of calculus it follows that

dU = TdS + SdT − pdV − V dp+∑i

µidNi +∑i

Nidµi

which when combined with Eq. 1 implies that

SdT − V dp+Nidµi = 0. (2)

This is called the Gibbs-Duhem relation. As a consequence of this relation, in a single component systemthe chemical potential µ is not independent of variations in Temperature and pressure.

Functions that express the intensive parameters, T, p and µi as functions of extensive parameters arecalled equations of state. For instance,

T = T (S, V,Ni)

is an equation of state. A single equation of state, unlike a fundamental relation, is not a complete descriptionof a thermodynamic system; although together, all the equations of states do completely describe the system.The difference between an equation of state and a fundamental relation is apparent for instance in that theformer lack the first order homogeneity property of fundamental relations. For an ideal, monatomic, gas theequation of states

p =23UV −1 and T =

23U(NR)−1, (3)

where R = 8.314 J (K mol)−1 is the universal gas constant, completely describes the system–per the Gibbs-Duhem relation an equation of state for the chemical potential is not necessary. The ideal gas law followsdirectly from the above as pV = NRT.

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2.2 Variables

2.3 Intensive, Extensive, and specific variables

In thermodynamics an important distinction exists between extensive quantities and intensive quantities.As we have shown intensive variables arise as partial derivatives of the extensive variables, and they donot depend on the amount of matter present. In addition extensive variables can be expressed in a per-massbasis, this gives them an intensive like quantity (no explicit dependence on mass), although they are different.Generally extensive variables are expressed in upper case, while lower case is reserved for intensive variablesand specific quantities—the exception is temperature.

2.3.1 Thermodynamic Coordinates

It proves possible, and useful, to replace the independent variables in the fundamental relations with in-tensive variables. To do so requires coordinate transform out of which arise the thermodynamic potentials.These potentials replace the energy, or entropy as the extensive function in the fundamental relation. TheHelmholtz potential, F = U − TS, arises from rewriting the fundamental relation for energy in terms oftemperature instead of entropy, the enthalpy H = U + PV arises from writing the fundamental relation forenergy in terms of pressure instead of volume, and the Gibbs potential, G = U +PV −TS, arises from thetransformation of the fundamental equation for energy that replaces the entropy by the temperature and thepressure by the volume. Hence the fundamental relation in terms of the enthalpy takes the form

H = H(S, p,Ni)

with V = (∂H/∂p)S,Ni .

2.3.2 Composite Systems

The atmosphere is a composite, multi-component system that we idealized as “dry” air and water in itsvarious phases. For the most part the dry air behaves like a diatomic ideal gas at “low” temperatures, forwhich the pre-factor in the equations of state (3) is 2/5. To denote properties specific to the dry air weintroduce the subscript d, so for instance Rd = 287.15 J (kg K)−1. is the gas constant (multiplied by themolecular weight of dry air) in contrast to Rv the gas constant for water vapor. From our thermodynamicpostulates, the fundamental relation for a composite system is additive over the subsystems; so for instance,for a dry-air, water-vapor, liquid-water mixture, H = Hd + Hv + Hl. Equilibrium between the phases ofwater vapor introduces a constraint between the phases (gv = gc) where here c denotes the condensatephase, either liquid or ice. So that a three component system of water vapor, condensate and dry air has thesame number of degrees of freedom as a two component system.

2.3.3 The many variables of atmospheric thermodynamics

Measuring the amount of matter The amount of matter, for instance water vapor, is measured in differentways in the atmosphere. Because studies of atmospheric physics are rarely concerned with chemistry thethermodynamic relations are often specified interms of constituent masses, or specific masses, rather thanmole fraction (nv) . The water can be measured in terms of the specific humidity, which we denote qv =mv/m or the mass mixing ratio, where the ratio of a constituent is referred to the amount of dry air, such thatrv = mv/md.Alternatively one can quantify the amount of vapor in terms of the vapor density, ρv = mv/v,or the vapor pressure, pv = ρvRvT which also depends on Temperature. Complementarily, given an ambient

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pressure, the amount of vapor in subsaturated air can also be uniquely specified in terms of a temperature,which is called the dew point temperature, Td, where Td is defined such that qs(Td, p) = qv. Finally wenote that the notation in atmospheric thermodynamics is not systematized, some cloud-physics texts referto mixing-ratios interms of w to avoid confusion with r which may be reserved for the radius, and vaporpressure is often denoted by the variable e.

Energy variables Temperature, T, is a measure of energy, this is implicit in the definition of the Boltz-mann constant, kB whose value of 1.3806 10−23 measures the ratio of Joules to Kelvin, so that kBT isformally recognized as an energy. The enthalpy is also an energy variable, as if a closed system is heatedwhile at a fixed pressure, the change in the enthalpy corresponds to the amount of heating, in a way that isanalogous to the change in internal energy that accompanies the heating of a system at fixed volume. Asmentioned earlier this encourages the use of the enthalpies because so many processes in the atmosphereoccur at a given pressure.

A variable related to temperature, but incorporating auxiliary information is the potential temperature,θ = T (p/p0)Rd/cpd defined as the temperature air would have if reversibly brought to a reference pressure,p0 which by convention is taken as 1000 hPa. The dry static energy, φ = h+gz, is analogous to the potentialtemperature except that it involves the additional assumption that pressure changes following the adiabatictransformation are hydrostatic. The liquid water, and equivalent potential temperature extend the conceptof reference temperatures to moist systems, in that they specify the temperature air would have if reversiblytransformed to a reference pressure and reference state of the water. In the case of

θl ≈ T (p/p0)Rd/cpd exp(−`vqlcpT

)this reference state is one in which all the condensate is evaporated. In this expression `v denotes theenthalpy of vaporization, namely `v = hv − hl. Hence θl can be thought of as an evaporation potentialtemperature. In contrast,

θe ≈ T (p/p0)Rd/cpd exp(`vqvcpT

)is a condensation potential temperature as it refers temperature to a reference state in which all the conden-sate is condensed. Hence θl ≤ θ ≤ θe. Analogous variables for the static energies are the liquid water staticenergy, φl = cpT − `vql + gz and the moist (equivalent) static energy, φe = cpT + `vqv + gz.

It has proven useful to introduce a great many other temperatures in the study of the atmosphere, amongthese are the virtual temperature, Tv = T (1 + (Rv/Rd− 1)qv− ql) whose fluctuations scale with buoyancyfluctuations, and derived variables, like the virtual potential temperature, θv = Tv(p/p0)Rd/cpd , or thesaturated equivalent potential temperature, θe,s wherein the water vapor in the definition of θe is replaced byits saturation value, making θe,s only a function of temperature and pressure; a static energy can be definedanalogously.

Overloading One of the difficulties of atmospheric thermodynamics is that there is no widely adheredto nomenclature. Thus certain symbols will be used to denote very different things in different contexts.Egregious is the symbol v which refers to the meridional velocity of air-currents, vapor, specific volume,or virtual effects depending on the context, and the symbol s which one finds used for specific entropy, drystatic energy, saturation, solute, or simply to denote a surface value.

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2.4 Processes

Of the varied thermodynamic processes that are studied in the atmosphere, by far the most important is theconcept of an isentropic process. For such a process, in a closed system, Eq. (1) simplifies to

dU + pdV = 0. (4)

Often we speak of adiabatic and isentropic processes interchangeably. The main difference is that irre-versible mixing within a multi-component system (say of warm and dry air) adds entropy to the system, butis not diabatic. In this case TdS > Q = 0, where Q is the heating.

2.5 Saturation

Figure 3: Saturation vapor pressure over liquid (dark grey) and ice (blue). At T = 0 C the saturationvapor pressure is 610.15 Pa. At T = −30 C the saturation vapor pressure over liquid water is 50.8 Paas compared to 38.0 Pa over ice at the same temperature. Saturation with respect to liquid at temperaturesbelow 0 C are relevant because super-cooled water is often present in the atmosphere.

The main purpose of all of the above is to introduce a language for speaking about saturation. Thedevelopment of saturation within the atmosphere is a necessary condition for cloud formation. From thegeneral condition of equilibrium systems containing two phases one can readily show that co-existencebetween two phases introduces a constraint on the system, which can be expressed as a relation betweensaturation pressure, es, and temperature,

d ln esdT

=`v

RvT 2, (5)

which is often referred to as the Clausius-Clapeyron equation; although more precisely it expresses theClapeyron relation under the Clausius approximation. A numerical approximation to the solution of (5)is given in Fig. 3 for saturation with respect to liquid, and equivalent expression for ice requires `v being

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replaced by the enthalpy of sublimation. The saturation condition is e = es(T ). The relative humidity, RH,is thus,

RH =e

es(T ). (6)

and one can speak of supersaturation, s as relative overabundance of water vapor, such that

s ≡ e− es(T )es(T )

= RH − 1. (7)

Air becomes supersaturated either by increasing the vapor pressure, or, because es is a strictly increasingfunction of temperature, by decreasing T. As a rule it is easier to bring air to saturation by cooling it than itis by moistening it, because RH is linear in the vapor temperature, but exponential in temperature.

3 Lecture 3: Droplet Activation

Of the thermodynamic potentials, the Gibbs Free Energy, proves convenient for exploring supersaturationin the atmosphere. The fundamental relation expressed in terms of the Gibbs potential, , G takes the form:

G = G(T, p,N) (8)

which given the Gibbs-Duhem relationship implies that for a single component system

G = SdT − V d p. (9)

From the main (Entropy) postulate of thermodynamics it follows that in equilibrium the thermodynamiccoordinates (T, p,N), take on values that maximize G. This maximum principle in G provides a basis forderiving the coexistence lines between two phases of a substance, in our case water. As a consequence ofthis maximum principle, chemical equilibrium between the two phases requires that gv = gl, and in turn,that for a change in the ostensible conditions of equilibrium, δgv = δgl. From (9) this constrains changes inpressure and temperature,

(sv − sl)dT = (vv − vl)dp, (10)

where thermal and mechanical equilibrium is assumed so that Tv = Tl = T, and pv = pl = p. Recognizingthat in equilibrium sv = sl = (hv−hl)/T = `v, leads directly to the Clapeyron equation, whose expressionin the form (5) requires one to additionally assume that the vl vv and that the vapor behaves as an idealgas.

It turns out that for this relationship to be useful in the study of clouds it has to be modified to accountfor two, in the end, decisive effects. The first is that saturated surfaces are curved, and so the surface energyassociated with surface tension must be accounted for, the second is that the liquid phase in the atmosphereis, for reasons we shall shortly explain, almost always a solution. These two effects compete. Surfacetension effects act to require a higher vapor pressure in equilibrium, solute effects require a lower vaporpressure in equilibrium.

Heuristically these effects can be rationalized with the help of Fig. 4. The left panel indicates thatmolecules are continuously changing phase, but that in equilibrium the rate at which particles are enteringthe condensed phase (condensation) equals the rate at which particles are leaving the condensed phase(evaporating), indicated by C and E respectively. In this example the condensed phase is shown to beseparated by a planar surface, but in actuality, this situation which is described by (5), makes no reference

11

EC

Figure 4: Equilibrium saturation vapor pressure between a liquid and a vapor for different situations: for asurface with no surface tension effects (right) ; for a curved surface with surface tension effects (middle);over a dilute solution, where solute is shown by darkened circles (right). The motion of only some of themolecules is indicated for the purposes of illustration.

to a surface whatsoever. In the second example the curvature of the surface and the molecular interactionsof molecules in this surface layer define a distinct surface phase with a surface energy that one has to workagainst to expand the condensed phase. Hence a greater vapor pressure is required for particles to enter thecondensed phase. Finally in the solution the depletion of the condensed phase at the phase boundary (dueto the present of solute) reduces the chance of evaporation, thus in equilibrium a smaller condensation rate(and hence a smaller vapor) pressure is required to balance the reduced rate of evaporation.

To account for these effects requires us to be quantitative, to do so requires the development of additionalconcepts.

3.1 Supersaturation over curved surfaces

In modeling surface effects it proves useful to think of the surface as a distinct phase. So doing implies adescription of the liquid, surface, vapor system as a composition over three subsystems. Each has its ownenergy, so that U = Uv + Uσ + Ul where the subscript σ denotes the surface subsystem. Similarly theentropy is given as S = Sv + Sσ + Sl, and the number of molecules is denoted N = Nv + Nσ + Nl. Thesystems differ in that the extensive description of the space occupied by the surface is denoted by its area, Ω,while that occupied by the bulk (vapor/liquid) phases is described by their volume. Hence the fundamentalrelation takes the form

U = Uv(Sv, Vv, Nv) + Uσ(Sσ,Ω, Nσ) + Ul(Sl, Vl, Nl). (11)

The differential of U is just,

dU = dUv + dUσ + dUl= Tv dSv + Tσ dSσ + Tl dSl − pv dVv + σ d Ω− pl dVl+ µv dNv + µσ dNσ + µl dNl. (12)

This expression implicitly introduces a new intensive variable,

σ ≡ ∂Uσ∂Ω

∣∣∣∣Sσ ,Nσ

(13)

which we call the surface tension. It has units of force per distance, or energy per area, in analogy to thepressure which is an energy per volume (or force per area). It is positive because surface is within thesystem, so work done on it adds energy to the system.

12

The entropy postulate implies that the state variables at equilibrium are those that maximize the energyof the system, so that in the present case, dU = 0. For a closed system, dS = dN = dV = 0, furtherassuming a surface, Ω, bounding a sphere of radius a d Ω = (2/a)dVl. it follows that

(Tv − Tσ) dSv + (Tl − Tσ) dSl −(pv + 2

σ

a− pl

)dVv + (µv − µσ) dNv + (µl − µσ) dNl = 0. (14)

The above can only be satisfied in general (i.e., for arbitrary variations) if the pre-factors themselves areidentically zero. Hence the conditions for thermal, chemical and mechanical equilibrium:

Tv = Tl = Tσ = T (15)

µv = µl = µσ = µ (16)

pv + 2σ

a= pl. (17)

To explore how the surface tension effects alter the Clapeyron equation we return to the condition forchemical equilibrium, which lead to Eq.(10). But this time we employ the mechanical equilibrium constraintin relating dpl to dpv, namely

dpv +2σa2

da = dpl. (18)

Substituting above yields the surface tension modified form of (10), namely

(sv − sl)dT = (vv − vl)d p−(

2σa2

)d a (19)

To isolate the effects of radius we specify T to some constant value and ask how p changes as we integratefrom an initial radius of infinity, where surface tension effects vanish by definition, to some desired radiusa. To emphasize that this relationship describes the saturation vapor pressure, rather than just any vaporpressure, we introduce standard atmospheric notation, wherein the saturated vapor pressure is denoted byes. The base value of es which is for pure water in the absence of surface tension effects is a function of Tonly, and it is denoted as e?.

With constant temperature, and in terms of es instead of pv (19) becomes

d es =−2σa2

(vlvv

)=−2σa2

(ρv

ρl

), (20)

so that

ln(es(a, T )e?

)=∫ a

∞

d eses

=∫ a

∞

−2σρlRvT

d aa2. (21)

which to the extent compressibility effects in the condensed phase can be neglected, and σ is independentof a, yields

es(T, a) = e?(T ) exp(

2σaρlRvT

). (22)

This is Kelvin’s equation, it demonstrates that the saturation vapor pressure increases strongly as a result ofthe surface energy barrier to condensation.

The saturation ratio, S ≡ es/e?, further simplifies notation where it is understood that S will differfrom unity to the extent curvature or solute effects change es. To avoid confusion note the slight distinction

13

between the symbols used for saturation ratio versus that for entropy, S and S respectively. In terms of thisnotation Kelvin’s equation becomes:

S(T, a) = exp(

2σaρlRvT

). (23)

An interesting implication of Kelvin’s equation is that for pure water droplets to be in equilibrium requiresextraordinarily high supersaturations, or the spontaneous generation of unusually large drops. This can beinterpreted as a barrier to homogenous nucleation, whereby we mean the production of droplets out of purevapor. Because of this barrier, the atmosphere almost always forms cloud droplets through heterogeneousnucleation, which takes advantage of the efficacy of solute in reduce the saturation vapor pressure over adrop, thereby counteracting the implied enhancement in the saturation vapor pressure due to surface tensioneffects.

3.2 Solute effects

The starting point for thinking about solution effects is Raoult’s law, whereby

p =∑i

= xip?i (24)

with xi denoting the mole fraction, xi = ni/∑

i ni and p?i denoting the saturation pressure in the absenceof solution effects, i.e., for a pure substance.

To be specific, consider the saturation vapor pressure over a solution of salt, NaCl, and water. Let usdenote by ns the number of moles of sodium chloride (NaCl) being dissolved in the water, and by nw thenumber of moles of liquid water. Raoult’s law states that the saturation vapor pressure will depend on themole fraction of H2O and temperature as follows

ew(x) = xwe? =(

nwnw + ns

)e? ≈

(1− ns

nw

)e?. (25)

The approximate form of the last equality is based on the assumption that ns nw. A solution that isdescribed by Raoult’s law is what we call an ideal solution, an analogy to the concept of an ideal gas. Forideal solutions the vapor pressure is reduced relative to what one finds for pure (undilute) solvents.

Experiments investigating these effects yield two conclusions. First, in the dilute limit the theory agreeswith measurements if the amount of solute is (in the case of NaCl) increased by a factor of two. Thisis taken as support for the idea that the solution disassociates into constituent ions, namely Na+ and Cl−

so that nsolute = 2ns. The second point is that even with this factor the agreement is only approximate,and that the adjusted theory can substantially depart from the measurements for even modest amounts ofsolute. These effects are thought of interms of departure of ideality. The fact that real solutions are not welldescribed by Raoult’s law has lead to a more empirical description based on the idea of an effective molefraction, aw which is called the water activity,

aw ≡ese?. (26)

Rather than directly tabulating the activity of different substances, departures from ideality are insteadfit by writing

aw = exp(−ν ns

nwΦs

), (27)

14

where ν is the dissassociativity (in the case of NaCl ν = 2) and Φs is the “practical osmotic coefficient” orif you like, a fitting factor, it is usually less than one. Note that aw is not just a function of the mole fractionof water, but also the type of solute. If ns nw and Φs = 1 then

aw = exp(−ν ns

nw

)≈ 1− ν ns

nw, (28)

which agrees with our expression (25) for Raoult’s law if one associated νns with the number of moles ofsolute.

Through the activity it proves straight forward to include solute effects in our expression for saturationvapor pressure. Noting that our original derivation to account for the effects of surface tension is independentof whether or not solute was included, in so far as we assume that the solute does not affect the surfacetension. In this case we can write

es(a)e?

=es(a;xw)es(∞;xw)

,

but es(∞, xw) = e?aw. From this it follows that

es(a, xw) = e?aw exp(

2σaρlRvT

)(29)

= e? exp(

2σaρlRvT

− ν nsnw

Φs

). (30)

Eq. 30 is known as the Kohler equation. It describes the saturation vapor pressure over a droplet solutionand is the basis for understanding both water active aerosol, which can deliquesce (grow) to large sizes ina humid environment, as well as the activation of cloud droplets which grow readily by diffusion in anenvironment that is supersaturated.

3.3 The Kohler equation and its properties

To explore the behavior of the Kohler equation for the supersaturation over a solution droplet, here werewrite it by noting that the relative salt and liquid mass on which it depends, can be rewritten in termsof the equivalent size of a solution droplet. To do so note that ns = ms/Ms, where ms and Ms are themass and molecular mass of the solute respectively, and nw = mw/Mw, with mw and Mw the mass andmolecular mass of H2O. The mass of the solute can be expressed in terms of an equivalent dry radius as sothat ms = (4/3)πa3

sρs while the mass of the water is given by mw = (4/3)π(a3−a3s)ρw. Combining these

two expressions and assuming that (3−a3s) ≈ a3 yields

nsnw

=a3sρsMl

a3ρlMs.

With this relation it is possible to express Kohlers equation in the somewhat simpler form

S(a,Bs, T ) = exp(A(T )a− Bsa3

). (31)

where A = 2σ/(ρlRvT ) is a weak function of temperature, and the solute is described by Bs in terms of itseffective size, as, and its chemical composition, such that

Bs = a3sχs where χs = ν(ρs/ρl)(Ml/Ms)Φs.

15

Eq. 31 emphasizes that the equilibrium saturation over a solution drop depends on the size of the drop, theamount of solute, and its chemical properties encapsulated by χs. For a given amount, and type, of soluteEq. (31) thus describes how the saturation vapor pressure over a solution drop changes as a function of thesize of drop.

For very small drops the solution effect, the term proportional to 1/a3 in Eq. (31) dominates, while forlarger drops the curvature term, A becomes more important.

Physically, having gone to so much work to derive an expresion for S it merits reflecting on what thisquantity describes. Quite simply, the saturation over a solution drop. If at some temperature T and for someamount and type of solute S is less than one, it implies that such solution drops can exist in equilibrium atrelative humidities equal to S. But the cubic form to the argument in the exponential form for S endows itwith physically interesting properties.

It is straight forward to show that S vanishes as a goes to zero, implying that even in a very dry atmo-sphere small solution drops can exist in equilibrium. Likewise as a becomes large S asymptotes to unity,showing that the largest drops require a supersaturated atmosphere. For an intermediate value of a the socalled critical radius, which is denoted ac, S takes on a maximum value, at which point its shape switchesfrom concave to convex. Drops larger than ac can be shown to be unstable, while those smaller than ac arestable. Physically this implies that in an environment with some fixed supersaturation, solution drops largerthan ac will grow unbounded, while those smaller than ac will remain at some small equilibrium size. Thisseparation of the population of solution drops, into the subset larger than a certain size that continue to grow,and a subset smaller than a certain size that do not grow, is the basis for our concept of a cloud, a dispersionof particles growing by vapor diffusion. In contrast to the aerosol, which we think of as small particles,often in solution, in equilibrium with the environmental humidity.

4 Lecture 4: Further Properties of an Isolated Drop

4.1 Diffusional growth

Droplet growth, or evaporation, is described by diffusional growth theory. Drops grow because they findthemselves in an environment where the saturation vapor pressure exceeds the equilibrium vapor pressure atthe surface of the droplet, likewise they shrink, if the saturation vapor pressure at their surface exceeds thatof their environment. The vapor pressure gradient determines whether or not water vapor molecules diffusetoward, or away, from the surface of droplets.

In most instances the drop radius is much larger than the mean free path of the water vapor molecules,and it is okay to treat condensation/evaporation using a continuum description encapsulated by the theory ofdiffusion. For very small droplets or haze particles it may become necessary to consider kinetic effects, i.e.,the statistics of molecular interactions described by the kinetic theory of gases, so as to modify the standarddiffusion-based description.

The starting point for the standard description of diffusional growth is Ficks Law in spherical coordi-nates, so that for a vapor concentration n,

∂n

∂t= D∇2n = ∇ · (D∇n) , (32)

where D is the diffusion coefficient of water vapor in air, which is constant, and n = n(R) is a function ofthe independent variable R > r which measures the distance from the center of a droplet of radius r, e.g.,Fig. 5 Note the change in notation from the last lecture where a denoted the droplet size. F

16

Figure 5: Schematic diagram showing vapor diffusing toward a growing drop of radius r. Here sphericalgeometry is assumed and the flux across a control volume of radius R is indicated by the thickened arrow.

Fick’s law states that the concentration gradient of the vapor is determined by the divergence of the vaporflux, Jn = D∇n, because the problem is spherically symmetric, a spherical coordinate system provides themost convenient description, in which case,

Jn = DdndR

. (33)

In steady state one expects the flux divergence to be zero, which implies that the flux of molecules across animaginary spherical surface of radius R is constant, i.e.,

R2Jn = Const. −→ dndR

=ConstR2

. (34)

Integrating it is straightforward to show that the general solution to this differential equation takes on theform

n(R) = C1 −C2

R.

Given the boundary conditions that n(R = ∞) = n∞ and limR→r n(R) = nr we can rewrite the generalsolution in terms of the concentrations of vapor molecules just beyond the surface of the drop and in the farfield, here taken as R =∞. So that the steady-state vapor concentration becomes

n(R) = n∞ − (n∞ − nr)r

R. (35)

Hence the steady state flux of water molecules across the surface R is just

4πR2Jd = 4πR2D(n∞ − nr)r

R2. (36)

These molecules accumulate on the surface of the growing drop so that the change in the mass of the dropis simply given by the flux of molecules to its surface, that is

dmdt

= 4πR2D(n∞ − nr)r

R2mH2O = 4πrD(ρv,∞ − ρs,r), (37)

17

where mH2O is the molecular mass of a water vapor molecule, and we write nrmH2O = ρs,r recognizingthat the vapor density at the drop surface is the saturation vapor density for that surface.

The important point in Eq. 37 is that diffusional growth theory predicts that droplets will grow whenthe vapor density away from the drop is larger than that near the drop and that this growth rate proportionalto the radius r of the drop and the diffusivity D. Note that the mass of a drop of radius r is m = 4π

3 r3ρl,

from which it follows that dm = 4πρlr2dr, which upon substitution into our diffusional growth law yieldsa description of how that radius of a drop grows for steady diffusional growth, namely,

drdt

=D

rρl(ρv,∞ − ρs,r). (38)

In terms of the growth of the radius of a drop growing through condensation the growth rate is inverselyproportional to its radius, indicating that while large drops accumulate more mass than small drops, theirradius changes less rapidly. From the perspective of radius, small drops grow faster.

With the help of the ideal gas law it is straightforward to write (38) in terms of the saturation vaporpressure ratio

drdt

=1r

D

RvTρl(e∞ − er) =

1r

De?RvTρl

[S − exp

(A(T )r− Bsr3

)]. (39)

The basic form of this equation is perhaps more clear if we define FD(T ) = RvTρlDe?(T ) to be a thermodynamic

factor, and note that for large enough drops the curvature and solute corrections to the saturation vaporpressure are negligible, so that

drdt

=1r

(S − 1FD

). (40)

Equation (39), or in a somewhat simplified form, Eq. (40), describes the basic physics of droplet diffu-sional growth. The growth rate is primarily determined by the degree of supersaturation and the droplet size,but modified by a thermodynamic function that encapsulates the diffusivity. Quantitatively this descriptionneglects three importan effects, two small effects are related to drop size, and one larger effect that is appar-ent for all drops. We will discuss these effects in turn, but in so doing note that none really fundamentallychanges the character of Eq. (40).

4.1.1 Temperature corrections

In deriving (39) we assumed that the temperature was constant everywhere. This is not a good assumption.As water vapor molecules condense on the surface of the drop their enthalpy of vaporization (latent heat) isabsorbed by the condensed phase which warms the growing droplet. To support the continued diffusion ofwater vapor molecules to the droplet surface requires enthalpy (heat) to be diffused in the other direction,away from the droplet into the surrounding media. Hence droplet growth depends simultaneously on thediffusivity of the medium of growth, but also its conductivity.

To account for the diffusion of enthalpy away from a growing drop (or toward an evaporating drop)requires us to solve the steady-state equation for thermal diffusion, which in analogy to the development ofEq. (37) takes the form

dHdt

= −4πrK(Tr − T∞), (41)

where H is the enthalpy and K is the thermal conductivity. It can be written interms of the thermal diffu-sively κ by noting that K = ρvcpvκ. For the enthalpy transport away from the drop to balance the accumu-

18

lation of enthalpy from condensation at the drop surface requires that

dHdt− λdm

dt= 0, (42)

or equivalently thatλD(ρ∞ − ρs,r) = K(Tr − T∞). (43)

Because ρv,r describes the saturation vapor density at the drops surface, and for a given drop size, soluteamount and type, it is only a function, Tr, the temperature at the surface of the drop, Eq. (43) is an implicitequation of Tr, or equivalently ρv,r. Its solution depends on the condensation rate and so it must be solvedsimultaneously with Eq. (38) when solving for the diffusional growth of the drop. Analytic solutions to thispair of coupled equations are not know, and they are customarily solved numerically.

It is, with a small approximation, possible to incorporate the heat-conduction effects into the dropletgrowth equation, to yield a modified form of Eq. (40) which will is not possible to solve analytically, but atleast more clearly illustrates the contraint imposed by the requirement that the drop balance the diffusion ofmass to or from its surface by the diffusion of heat in the opposite direction. The approximation we make isto expand ρv,r in a Taylor series about the environmental temperature, T∞, such that

ρv,r(Tr) = ρv,r(T∞) +d

dT

∣∣∣∣T∞

(Tr − T ). (44)

But (Tr − T ) can be expressed in terms of the condensation rate (from Eqs. 41-42) such that

ρv,r(Tr) = ρv,r(T∞) +dρs,rdT

∣∣∣∣T∞

(λ

4πrKdmdT

). (45)

Substituting this expression for ρs,r in Eq. 37 and rearranging yields

dmdt

= 4πrD

ρv,∞ − ρs,r(T∞)

1 + DλK

dρs,rdT

∣∣∣T∞

, (46)

which is the desired expression for the growth of a droplet of radius r in a vapor field whose density is givenby ρv,∞ and whose temperature is denoted by T∞.

In the large drop limit, where we can neglect curvature and solute effects on ρs,r we note that

dρs,rdT

∣∣∣∣T∞

=1

RvT∞

ddT

e?(T∞)− e?(T∞)RvT 2

∞=

e?RvT 2

∞

(λ

RvT∞− 1). (47)

Substituting above allows us to write a growth equation which is analogous to Eq. (40), but which incorpo-rates temperature effects, namely

drdt

=1r

(S − 1

FD + FK

), (48)

with

FK(T ) =λρlKT

(λ

RvT− 1)

and FD(T ) =RvTρlDe?(T )

. (49)

This is the form of the droplet growth equation that is most commonly used to describe the growth of clouddrops after they have been activated. The conductivity related thermodynamic factor, FK is of approximatelythe same magnitude as the diffusivity factor FD which implies that the growth of drops is reduced by abouta factor of two due to the finite conductivity constraint.

19

4.1.2 Drop size effects on droplet growth

The effect of drop size modifies not only the saturation vapor pressure at the drops surface but also theeffective diffusivity in the droplet near field. There are two categories of effects. The first is related tocorrections to the diffusivity that emerge for very small drops, where the continuum descriptions ceases to bea good one, and must be corrected for molecular kinetic effects. The second is when the drop is sufficientlylarge to fall through the vapor field, thus distorting the equilibrium vapor field in the vicinity of the dropletsurface, it is often called the ventilation effect. The kinetic effects reduces the effective diffusivity andconductivity and thus retards the growth of the drop, the ventilation effects enhances the effective diffusivityand conductivity thereby enhancing the growth (or in the case of evaporation, shrinkage) of the drop.

Kinetic Effects: Kinetic corrections are applied by using an effective diffusivity, D′ = Dg(β) and aneffective conductivity K ′ = Kf(α) in the droplet growth equations. The functions g(β) and f(α) aresemi-empirical corrections that take the form

g(β) =r

r + κβvβ

and f(α) =r

r + Dαvα

. (50)

where recall that κ defines the thermal diffusivity, so that κ/(βvβ), with vβ a molecular velocity scale thatmay depends on temperature, defines an effective length-scale. The expression for f(α) is defined similarly.The coefficients β and α are called the condensation and accommodation coefficients respectively and arepoorly known. Standard texts take α ≈ 0.05 and β ≈ 1. The form of g(β) and f(α) guarantees thatD′ < Dand K ′ < K hence the corrections have the effect of slowing the growth of small drops.

Ventilation Effects: When drops are large they fall through the ambient medium with an appreciablevelocity. This disturbs the vapor field that attempts to equilibrate around the surface of the drop through theprocess of steady diffusional growth effectively enhancing the effective diffusivity and conductivity of themedium, not unlike the wind enhances evaporation from the surface. This enhancement factor is usuallydenoted by fv and is described empirically, as a function of the Reynolds number, Re, here defined as

Re =ru∞,rν

(51)

, where u∞,r, denotes the terminal velocity of a drop of size r and ν (not to be confused with the dis-sassociativity in the previous lecture) is the dynamic viscosity of the fluid through which the drop ismoving. The ventilation correction takes the same form for both the diffusivity and conductivity, so thatD′/D = K ′/K = fv > 1 :

fv =

1.00 + 0.09Re 0 ≤ Re ≤ 1.250.78 + 0.28Re1/2 Re > 1.25

(52)

For large-drops the Reynolds number can be greater than 100, this ventilation effect can significantly en-hance the rate of evaporation and condensation of precipitation sized particles.

4.2 Terminal fall speeds of drops and droplets

The rate at which drops fall through air depends both on the size of the drop and the density of air. Theequilibrium fall speed of a drop can be found by equating its acceleration with the drat it experiences. The

20

latter depends on the fall speed of the drop thus yield the terminal velocity, i.e., that velocity at which thedrag exactly balances the gravitational acceleration.

The gravitational acceleration we denote by m′g where m′ is the effective mass of the drop, i.e., itsmass minus the mass of the fluid it displaces. The drag is more complicated, but should, on the basis ofdimensional arguments, take the form

D = r2ρu2∞,rf(Re), (53)

which implies that

u∞,r =(

4πr3f(Re)

(ρl/ρ− 1))1/2

≈(

4πrρl3ρf(Re)

)1/2

(54)

Where the approximation is justified, because ρl ρ typically we neglect the factor of unity relative toρl/ρ. However Eq. 54 is misleading, and indeterminate, because the Reynolds number, is itself a functionof u∞,r.

Unfortunately, even for solid spheres, is has not been possible to derive expressions for the drag theyexperience when moving through an ambient fluid, and except for a very limited regime,D must be obtainedempirically. The limited regime where analytic solutions can be found is appropriate for small dropletsmoving through a laminar fluid for which the pressure gradients and viscosity balance the gravitationalacceleration of the drop. In this regime, which we call Stokes flow, Re 1, and the terminal velocity thatdevelops is referred to as the Stokes velocity. The drag on a sphere in Stokes flow can be derived exactly as

DS = 6πrνu∞,r (55)

from which it follows that

u∞,r = US =2ρlr2

9ρµfor Re 1. (56)

This expression provides the important result for that the great variety of small particulate matter the fallspeed varies as radius of the particle squared, and only when the particles reach an appreciable size do theybegin to fall with an appreciable velocity relative to the variable air currents in which they are embedded.

A considerable effort has been devoted to extending Stokes solution to consider effects that emerge atlarger Reynolds number, namely the development of turbulent fluid boundary layers at the drops surface, orwakes behind the drops. And while some improvements have been derived for values of Re near unity, forthe most part our description of the fall speed of drops is empirical, based on the fitting of functional formsto data collected in the laboratory. Typically this data is organized into different regimes. For small droplets,typically less than 20 µm in diameter, the Stokes solutions provides a good description, for intermediate sizedrops whose diameter range from a few tens of microns to perhaps a millimeter the fall velocity increasesless rapidly with r, and for very large drops a variety of other effects come into play, including the surfacetension at the surface of the drop, changes in the shape of the drop and perhaps even internal circulationswithin the drop, all of which combine to yield a very weak dependence of the drop fall velocity on the sizeof the drop itself for drops larger than about 1 mm in diameter.

One expression that has been developed which gives a reasonably good fit to the empirical data, and israther easy to calculate is based on the following expression

u∞(r) = exp(y) where y =m∑j=0

cj (ln(2r))j (57)

and cj depending on the size of the drop as indicated in Table 1

21

r ≤ 40µm r > 40µmc0 10.5035 c0 6.5639c1 1.08750 c1 -1.0391c2 -0.133245 c2 -1.4001c3 -0.00659969 c3 -0.82736

c4 -0.34277c5 -0.083072c6 -0.010583c7 -0.00054208

Table 1: Coefficients for Eq. 57 describing the terminal velocity of droplets in two regimes. These are takenfrom Table 4 of Beard (1977) and are valid for saturated air at 20 degC and 1000 hPa. Fits based on thesedata differ from experimental data by less than 1% on average, with the fits for the small droplet range beingconsiderably more accurate.

5 Lecture 5: Populations of Particles

When discussing clouds it is important to consider not just one particle, but collections of particles. Asystem of many particles is interesting for two main reasons. Particles compete with one another for theavailable vapor, and particles can interact. Both effects are important for the description of clouds.

To describe populations of particles we need to develop a language for doing so. This is the language ofdistributions, or distribution functions. When speaking of a distribution one needs to know two things, whatis being distributed, and over what it is being distributed. For example: among a population of particles wemay be interested in how the total number of particles is distributed as a function of the diameter, D, of theparticles. In this case we can speak of the number, N, being the dependent variable, and the diameter Dbeing the independent variable over which the number is distributed. N is clearly a function of D. It provesuseful to be somewhat more precise and adopt the convention that

if N(D) is the number of particles whose diameter is less than or equal to D then N(D),then we can speak of N(D) being a distribution function,

in this case one that describes the distribution of particle number as a function of their diameter. In thepresent example there is nothing magical aboutN andD, one could choose other independent and dependentvariables. For instance one could ask how the mass, M, of particulate matter is distributed as a function ofdiameter. In this case M(D) would describe the mass distribution as a function of diameter. It sometimesproves useful to distribute a variable over the logarithm of one of its physical properties, for instance byasking how the number of particles is distributed as a function of the logarithm of particle mass, in whichcase N(lnD) describes how N is distributed over lnm.

From a mathematical point of view distribution functions are restricted to a class of functions withspecific mathematical properties. They are: (i) real valued; (ii) defined on R; (iii) are non-decreasing;(iv) and right continuous. In the present context points (i) and (ii) are self-evident. Point (iv) allows one toextend the concept of distribution functions to the discrete case, but is not important as in the present contextfunctions we will consider will anyway be continuous. Thus for us the mathematical content is embodied inthe second point, which reflects the cumulative nature of the distribution, per our definition above.

22

Given a distribution function it is straightforward to define a density function as

n(D) ≡ dNdD

. (58)

Per this definition n(D)dD is the number of particles in the infinitesimal size interval (D,D + dD), alter-natively:

n(D) = limδD→0

N(D + δD)−N(D)δD

. (59)

From the properties of the anti-derivative it follows that

N(D) =∫ D

−∞n(D′) dD ==

∫ D

0n(D′) dD, (60)

where the second equality follows in our case because n(D) = 0 for D ≤ 0. At this point the readermay have noticed that the distribution function, N(D) as we have defined it, differs from a cumulativedistribution function only by a normalization constant,N(∞). Consequently p(D) = n(D)/N(∞) satisfiesthe requirement of a probability density function in that

∫∞−∞ p(D) dD = 1.

5.1 Converting distribution and density functions

002001020

1

2

3

4

5

parti

cle

mas

s [d

M/d

(logD

)]

diameter [nm]00500200102

0.0

0.5

1.0

1.5

2.0

2.5

3.0

parti

cle

num

ber

[dN

/d(lo

gD)]

diameter [nm]

Figure 6: Submicron aerosol size distributions, raw data measured at Ragged Point Barbados in November2010. Data and plots courtesy of Heike Wex of the Institute for Tropospheric Research in Leipzig.

Given the density function n(D) it may be desirable to relate this density function to other densityfunctions. For instance, if the relationship between the diameter and the mass of a particle is known thengiven n(D) it is straight forward to solve for the distribution of particle mass, m(D). Here we recall that

m(D) ≡ dMdD

; (61)

however, the cumulative mass of particles whose diameter less than or equal to D is just

M(D) =∫ D

−∞(π/6)D′3ρln(D) dD, (62)

23

from which it follows that m(D) = (π/6)D3ρln(D). How one describes the data can have a bit influ-ence on the interpretation. In Fig. 6 it is quite clear that particles less than 0.1 µm dominate the numberconcentrations but particles larger than 0.1 µm in diameter dominate the mass distribution.

Alternatively, given n(D), one may ask about the density n(m). But from the definition of the densities(60) the number of particles in a size interval must be the same irrespective of how you measure that interval,so that

n(m)dm = n(D)dD, (63)

hencen(m) =

2πD2ρl

n(D). (64)

Plotting n(D) versus D provides a useful way to show how particles are distributed with size, particu-larly when the distribution does not take a shape that can readily be described in terms of an analytic functionof a few parameters. The area under the curve of n(D) plotted versus D givens the total number of drops;hence, plotting density functions on the same graph also is a useful basis for comparison. Because the rangeof particles can be very large, from nanometers to microns or larger, n(D) is often plotted versus log10(D),so as to give the different size ranges equal weight in the display of n(D). However doing so can concealsome of the information of the plot, for instance the area under the graph of n(D) plotted versus log10(D)is no longer N the total number of particles. This is readily evident by noting that∫

n(D) d(log10D) =∫n(D)

1D ln(10)

dD 6=∫n(D) dD. (65)

For this reason some authors prefer to plot n`(D) = n(D)D ln 10 because∫ D

−∞n`(D′) d(log10D

′) = N(D). (66)

To avoid confusion about what is being plotted, the convention has developed to denote the density functionin terms of its distribution function, i.e., n`(D) is written as dN(D)/d(log10D), while n(D) would bewritten as dN(D)/dD, and so on.

5.2 Time derivatives of densities

Given a description of the particles in terms of number, or mass density functions, our interests shift fromdescribing how a particle’s size evolves in time, to how a distribution, or density function involves in time.To do so consider a subset of particles between two sizes, D1 and D2. In this case we can write,

N(D2, t)−N(D1, t) =∫ D2

D1

n(D, t) dD (67)

where we have included time (denoted by t) as a dependent variable to describe the possible evolution ofthe distribution in time. From elementary calculus we note that

ddt

∫ D2

D1

n(D, t) dD =∫ D2

D1

∂n(D, t)∂t

dD −[

dD2

dtn(D2, t)−

dD1

dtn(D1, t)

](68)

=∫ D2

D1

∂n(D, t)∂t

+∂

∂r

(dDdtn(D, t)

)dD. (69)

24

The first equality is given by Hopital’s rule, the second from the fundamental theorem of calculus and thedefinition of the anti-derivative.

In the absence of a source or sink of new particles, but allowing for particles to grow or shrink with time,say as a result of condensation or evaporation, Eq. 69 implies that

∂n(D, t)∂t

+∂

∂r

(dDdtn(D, t)

)= 0. (70)

Here the time derivative dD/dt can be thought of as the velocity of the distribution as it moves throughdiameter space, which is given by the growth rate equation we derived for a single particle in Lecture 4,e.g., Eq. 48. More generally, n may also be a function of space, so that it describes the density of particlesper unit volume per unit size, in which case given a general source or sink, χn the above generalizes to thefollowing Equation for the population density of particles,

∂n

∂t+∇ · (vn) +

∂

∂r

(dDdtn

)= χn, (71)

where v is the vector velocity field, and (∇·) denotes the divergence.

5.3 Common distributions and their density functions

For many purposes, including simplified cloud models, the distribution of aerosol particles, droplets ordrops, can be (or are anyway) described by functions of a few parameters. A famous example is ChristianJunge1 who in studying small (Aitken) aerosol particles wrote: “For simplicity, the line spectra of the Aitkenparticles were converted to continuous distributions.” Junge described the continuous distributions as,

dNd log r

= cr−3 for particles greater than 0.1 µm in radius. (72)

This distribution has come to be known as the Junge distribution, it has the form of what is more generallycalled a power-law. Power law distributions have the property of appearing as a straight line on a log-logplot. However, because integrals of a power-law diverge, they tend to be only valid over a limited range, forinstance it is generally accepted that the Junge distribution is an increasingly poor description of particleslarger than 10 µm.

A slightly more general distribution that proves useful in the description of cloud and aerosol particlesis the exponential distribution. This probability density function for this distribution is given as

n(D) = Nλ exp(−λD), (73)

where N denotes the total number of particles over the entire size range, and λ is that distribution param-eter. Marshall and Palmer, in a paper in 1948 showed that rain-drops are well described by an exponentialdistribution, with the restriction that λ = 41R−0.21, and Nλ = 0.08cm−4, with R denoting the rain-rate inmm h−1. Such a description leads to a density function for raindrops that is only a function of the rain-rateR, making it very convenient, if not extraordinarily accurate.

1Christian Junge (1912-1996) was born in Elmshorn. He studied and worked in Hamburg and Frankfurt, and publishing hisJunge distribution as part of his habilitation research while at the University of Frankfurt. After a period of research in the UnitedStates Junge returned to take a professorship at the University of Mainz, and thereafter the directorship of the Max Planck Institutefor Chemistry. Junge is also well known for his discovery of the stratospheric aerosol layer sometimes called the Junge layer.

25

The Gamma distribution is a generalized form of the exponential distribution, and can be written interms of the number density function,

n(D;α, λ) = Nλα

Γ(α)Dα−1 exp(−λD), for D ≥ 0 (74)

where α and λ are the parameters of the distribution, and Γ denotes the Gamma function from whichthe Gamma-distribution derives its name. The exponential distribution is the special case of the Gammadistribution when α = 1.

A similar two parameter distribution function is the log-normal distribution function, whose number-density function takes the form

n(D;µ, σ) =1

Dσ√

2πexp

[− (lnD − µ)2

2σ2

]for D > 0. (75)

The log-normal distribution has the property that its geometric mean is given by exp(µ) while its geometricstandard deviation is given by exp(σ).

In practice, when fitting data to a particular distribution function it is important to take into account thefinite range of the data, so that the parameters of the distribution that best fit the data are chosen based ontruncated distributions, i.e., distributions defined over a finite range.

6 Lecture 6: Collection through Collision and Coalescence

Because cloud-base rarely extends to the surface, for rain to reach the surface it must generally fall throughsubsaturated air. In this regard the thought experiment of a small drop falling through cloud base in asubsaturated environment is intuitive. For a given relative humidity, evaporation is mostly a matter of time,and for smaller drops the settling time increases with the square of the drop radius, which thus allows moretime for evaporation and further reduces the drop radius. Clearly large-drops are required for precipitationto reach the surface as precipitation.

However it is not possible to grow rain-drops by condensation alone. A cloud whose maximum watercontent is 2 gm−3 would consist of drops of only about 30 µm in diameter given a concentration of 100cm−3. Given that the maximum water content of most clouds is less than 2 gm−3 and the droplet concen-trations somewhat higher, the chance of growing precipitation size droplets is even further reduced. Even ifrandom processes favored a very small percentage of the droplets to grow very large by condensation, a 1mm drop requires a 0.1 m3 volume of air whose adiabatic liquid water content is 5 g m−3 to be void of anyother drops, an effectively impossible requirement.

Empirically it is also readily apparent that rain-drops do not form by condensation alone. Observedsize distributions of rain drops suggest that the number concentration of raindrops is less than one per litre,which is 100,000 fold less than the concentration of cloud droplets. Both lines of thought suggest that rain,at least in worm clouds, requires cloud droplets to become aggregated into larger rain drops. This processof aggregation involves two steps: collisions between droplets and their subsequent coalescence. This twostep processes of aggregation is generally referred to as collection to emphasize the role of the larger dropin initiating the process, a point that should be come clearer in the subsequent discussion.

Collection is conceptually simple, and illustrated by the process of gravitational collection in the nextsection. Elaborations of this idea which allow one to more completely describe droplet interactions ismade possible by the theory of stochastic collection, whose mean-field equation was derived by MarinSmoluchowski in 1916 and is used in a wide number of fields.

26

6.1 Gravitational collection

The basic idea of gravitational collection is familiar to anyone who has watched raindrops slide down awindow pane. Larger drops slide faster, overtaking and assimilating the smaller drops that lie in their path.Mathematically this process can be described by asking how many drops fall in the path of the larger dropin some time interval ∆t.

R

r

[V(R) - v(r)] ∆t

Figure 7: Schematic illustrating the interaction volume for gravitational collection in which a collector dropof radius R falls through a volume of smaller drops, of radius r, some of which will be collected.

The basic set-up of this problem is shown in Fig. 7, where we envisage a large collector drop of radiusRand terminal, U∞ falling through a volume containing many smaller drops of radius r and terminal velocity,u∞. In some time interval ∆t the large-drop will meet all droplets in the cylinder whose volume is given by(R+ r)2‖U∞ − u∞‖. Assuming for simplicity that all collisions result in coalescence, then the mass of thelarge-drop will increase by the mass of all the small droplets it assimilates, such that

∆M = Nrπ(R+ r)2‖U∞ − u∞‖∆t(

43πr3ρl

).

Here the term in the brackets is the mass of a droplet of radius r and the pre-factor describes the number ofdrops colliding with the collector drop in time-interval ∆t. Taking the limit as ∆t goes to zero yields thegravitational collection equation, describing the rate of mass growth of a large-collector drop falling througha field of small cloud droplets:

dMdt

= Nrπ(R+ r)2‖U∞ − u∞‖(

43πr3ρl

). (76)

27

The gravitational collection equation can be readily generalize to allow for a population of cloud droplets,so that Nr = n(r)dr, and through the introduction of a collection efficiency, Ec(R, r), for the possibilitythat the collection volume is modified by flow, or other, effects, so that

dMdt

=∫ (

43πr3ρl

)K(R, r)n(r) dr, (77)

where we have introduced the collision Kernel,

K(R, r) = π(R+ r)2‖U∞(R)− u∞(r)‖Ec(R, r) (78)

to encapsulate the rate, per concentration of collected drops, at which a collector drop of radius R collideswith a droplet of radius r. It turns out that the collection efficiency, Ec(R, r), plays a non-trivial role in thetheory, as will be discussed subsequently.

6.2 Smoluchowski (stochastic) collection equation

The theory of gravitational collection presumes the existence of two populations of droplets: large collectordrops of radius, R, and smaller droplets, or radius r, which are then collected as a result of hydrodynamicalinteractions. The smaller droplets may be described by a distribution, but it is presumed that the larges ofthese smaller droplets is always much smaller than the collector drops, otherwise the possibility exists thatthey themselves might start actively collecting other drops.

This separation into collector, and collected drops, turns out to be an unnecessary idealization. In princi-ple any two drops can collide with some probability, and a less restrictive way to phrase the question is howdoes a population of droplets evolve as a result of such interactions. More precisely, now does the numberof drops of some volume x evolve in time if such interactions are allowed for. Denoting the number of dropsin some vanishingly small size interval (x, x + dx), at some time t, by n(x, t)dx, we are interested in thetemporal evolution of the distribution, namely,

∂

∂t[n(x, t)dx] .

Drops with a volume x can be created by binary interactions between two smaller drops. The coales-cence between a drop of mass x−y and a drop of mass y will create a drop of mass x. If we associated x−ywith the mass of the larger drop, then y can vary between 0 and x/2, and integrating over all interactionsyields the change in the number of drops of size x in a time interval ∆t, namely

∆ [n(x, t) dx] = ∆t

[dx∫ x/2

0n(x− y, t)n(y, t)K(x− y, y) dy

]. (79)

Here K is a generalized collection Kernel. It can be thought of measuring the interaction volume, modi-fied by the probability that colliding drops will indeed coalesce. In this case K(x, y)n(y)dy describes theprobability that a drop of mass x will collect a drop of mass y in the time interval ∆t.

Likewise, the number of drops of size x will be reduced each time a drop of size x collects another drop.From the definition of K the reduction of drops of size x in a time interval ∆t is thus

∆ [n(x, t) dx] = −∆tn(x, t) dx∫ ∞

0n(y, t)K(x, y) dy. (80)

28

As a result the net change in drops of size x is thus

∆ [n(x, t) dx] = ∆tdx

[∫ x/2

0n(x− y, t)n(y, t)K(x− y, y) dy − n(x, t)

∫ ∞0

n(y, t)K(x, y) dy

].

(81)Dividing both size by dx∆t and taking the limit as ∆t → 0 yields an integro-differential equation thatdescribes the evolution of the number distribution of drops as a result of binary interactions:

∂

∂tn(x, t) =

∫ x/2

0n(x− y, t)n(y, t)K(x− y, y) dy − n(x, t)

∫ ∞0

n(y, t)K(x, y) dy. (82)

This equation is sometimes called the Smoluchowski coalescence equation after Marian Smoluchowski whofirst derived it. By symmetry∫ x/2

0n(x− y, t)n(y, t)K(x− y, y) dy =

∫ x

x/2n(x− y, t)n(y, t)K(x− y, y) dy.

Hence, the Smoluchowski equation can also be written as

∂

∂tn(x, t) =

12

∫ x

0n(x− y, t)n(y, t)K(x− y, y) dy − n(x, t)

∫ ∞0

n(y, t)K(x, y) dy, (83)

which is the more usual form of its presentation.While Eq. 83 performs the book-keeping required to evolve a distribution of drops in time as a result

of binary interactions among the drops, the physics of the interaction is contained in the kernel, K. In thecloud-physics literature Eq. 83 is often called the Stochastic-Collection Equation (SCE), although it is apurely deterministic equation, i.e., it contains no stochastic element. The reason for this is because Eq. 83can be interpreted in as the mean-field representation of a stochastic process, analogous to the way diffusionis the mean field representation of Brownian motion; the latter being a stochastic process whose net effectcan be described deterministically.

6.3 The collection kernel

The kernel, K(x, y) encapsulates the physics of binary interactions among a population of droplets. Inprinciple it encapsulates all factors that regulate whether drops will collide and coalesce, thereby collectingone another. It makes sense to distinguish between collisions and coalescence because collisions need notresult in coalescence. Small drops may rebound from larger drops, and large-drops may be broken apart asa result of collisions. Moreover the collection kernel in the Smoluchowski equation can embody arbitraryphysical interactions, for instance coagulation as a result of collisions induced by brownian motion, whichare important for sub-micron particles.

For warm-rain formation, hydrodynamic interactions in the presence of a gravitational field are thoughtto underly the collection process. The relevant kernel is then given by Eq. 78 which is sometimes calledthe gravitational collection kernel. For this kernel collisions are brought about by differences in dropletsettling velocities modified by a collection efficiency which in principle accounts for effects modifying thecollisional cross section between two drops, as well as their coalescence efficiency. The hydrodynamickernel is the starting point for more advanced treatments of collision and coalescence, and hence its basicproperties merit discussion.

29

0.2 0.4 0.6 0.8 1.0

0.01

0.10

1.00

0.001

20 µm

10 µm

40 µm

r’/r

Collision Efficiency

Figure 8: Schematic to show the main features of different theoretical estimates of the collisional efficiencies(abstracted from Fig.14-5 of Pruppacher and Klett).

Important properties of the gravitational collection kernel are that it forbids drops of equal size collectingone another, and that it increases markedly with the size of the interacting drops. The first point is clear fromthe terminal velocity dependance of the Kenrel, wherein the difference between the terminal velocities oftwo equally sized droplets, and hence K vanishes. The second property is evident by noting that for smalldroplets the terminal velocity is proportional to r2, where r is the droplet radius. Hence, taking r′ < r

K(r, r′) ∝ (r + r′)2(r2 − r′2) = (1 + r′/r)2(1− (r′/r)2)r4

which shows that the collection rate increases not only with the difference in the size of the interacting drops,but also with the absolute size of the collecting drops—the latter being more important. Larger drops sweepout a larger volume and hence are more efficient collectors.

The above discussion does not account for how drops, when they fall, modify the flow field aroundthem, which in turn mediates their interactions with one another, and the effect of inertia, charges, andsurface properties which may cause colliding drops to coalesce, rebound or even disintegrate into manysmaller fragments. Hence the rate of collection given by the base kernel above must be a modified by anefficiency which incorporates many different effects. The one that has been studied in most detail is thecollision efficiency, namely modifications to the collisional cross section that result from hydrodynamiceffects. Fig. 8 illustrates the collision efficiency as understood based on theoretical/numerical studies. Themain points of Fig. 8 are that for small droplets (radius smaller than 40 µm) the collision efficiency is largestfor size ratios of about 0.4 to 0.8, and that the collision efficiency increases dramatically with size, roughlyin proportion to the square or cube of the collector droplet radius. The data shows some tendency of like-sized drops to be super-efficient, which is associated with wake-capture effects, wherein the interaction ofthe droplet wakes induces a collision that would not have otherwise been expected.

Generally the collection efficiency should be taken as the product of the collision efficiency and the co-alescence efficiency, however the former is uncertain to within perhaps a factor of two, and only qualitativeinformation about the latter is know, hence it is often assumed that uncertainties in the collision efficiencysubsume the effects that might cause the coalescence efficiency to depart from unity. Indeed present research

30

suggests that the largest gap in our understanding of the collection efficiency is not due to a poor understand-ing of coalescence efficiencies, but rather in how turbulence effects modify the collisional efficiencies.

If collision efficiencies scale with the square of the droplet radius then, at least for small droplets, thecollection Kernel as a whole, K, scales with r6, equivalently x2 where x denotes the mass. In practicethe collisional Kernel is constructed by interpolating between fixed points given by experimental data. Ithas however been found that a reasonable fit to this data can be obtained using polynomial functions. Forinstance Long suggests the following approximation to the Kernel:

K(x, y) =

9.44× 109(x2 + y2), max(x, y) ≤ 50µm5.78× 103(x+ y), max(x, y) > 50µm

. (84)

In Long’s fit to the experimental data we see that indeed the collision efficiency increases with the squareof the droplet mass for small, cloud droplets. An interesting aspect of the Long Kernel is that it does notretain the property of the gravitational kernel wherein a mono-disperse droplet spectrum will not grow bycollection, i.e., the kernel does not vanish when x = y. Instead the Long kernel emphasizes the role of largerdrops in accelerating the collection process, through its dependence on the square of the droplet mass.

7 Lecture 7: Warm clouds and drop spectral evolution

The discussion of the last lecture emphasizes the important role of the largest drop in the spectrum of clouddroplets. Exactly how large the largest drops must become before they effectively seed the coalescenceprocess is a matter of debate, however observations suggest that once the mean volume radius of the dropletspectrum reaches a size between 15 and 20 µm rain begins to develop efficiently in clouds. This summaryof the observations presumes, to some extent, the shape of the droplet spectrum. A broad droplet spectrumwhose mean radius is 15 µm may contain many more 30 µm drops than a narrow distribution of drops whosemean radius is 20 µm. The key point is to have a sufficient population of collector drops, or precipitationembryos.

The origin of collector drops, and hence a theory of warm rain, is a topic of active research. A numberof theories have been proposed, and generally can be cast into two categories: endogenous and exogenous.Endogenous theories invoke turbulence associated with cloud development in one of a number of fashions,exogenous theories make use of pre-existing conditions, for instance the presence of unusually large con-densation nuclei (so called giant or ultra-giant CCN), or preconditioning of the droplet spectrum throughprevious convection, as might happen if a new cloud rises through the remnants of a previous cloud.

7.1 Exogenous theories of warm rain

7.1.1 Giant CCN

Of the exogenous theories, the idea that giant or ultra-giant CCN can initiate the precipitation process comesup the most often. It is common to classified aerosol particles by sizes. To emphasize that they are ratherunusual, the term giant CCN has been invoked to describe CCN whose dry radius, rd falls in the rangebetween 1 and 10 µm, the term ultra-giant is reserved for particles whose dry radius is greater than 10 µm.For reference the radius of a salt particle at a humidity of 100% increases as r3/2d . So given that a salt particlewhose dry radius is 0.1 µm can grow to a haze particle whose size is 0.3 µm at 100 % relative humidity, asalt particle with a 2 µm dry radius will grow to about 25 µm at 100 % supersaturation.

The sea-surface is a ready source of large salt particles. Agitation of the water surface releases sea-spray into the air, and mixes air into the water column. It turns out that the mixing of air into the water

31

is more important for the production of sea-salt particles than the release of, and subsequent evaporationof sea-spray torn from the crests of waves. The air in the water column nucleates air bubbles that thentravel upward eventually reaching the surface and bursting. This bursting is due to the breaking of the watersurface as the bubble pushes upward, and it releases a fine spray that can be launched several centimetersabove the surface. A portion of this spray evaporates leaving behind sea-salt particles that are small enoughto remain suspended in the air, and be carried away from the surface. This phenomena is familiar to anyonewho has experienced a carbonated drink.

The production of sea-salt aerosol increases with wind speed, and can produce appreciable concentra-tions of giant aerosol particles at even modest, 10 m s−1 wind speeds. Observations of cumulus clouds,suggest that the height at which precipitation initiates in a cloud is independent of the number concentrationof cloud droplets. This result can be reconciled with our theoretical understanding of collision and coales-cence if a fixed number of giant CCN are assumed to be present irrespective of the ambient cloud dropletconcentration. While this result suggests that giant CCN may be important to the initial formation of rain inwarm clouds, it remains unclear how important giant or ultra giant CCN are to the subsequent developmentof rain showers, and the overall production of rain by shallow clouds.

7.1.2 Drop spectral preconditioning

Shallow maritime cumulus clouds in which the rapid formation of warm rain is something of a puzzlerarely develop out of the clear blue sky. Rather, convective pulses often appear along subtle boundariesin marine air-masses, perhaps associated with cold pools from previous convection, or in regions wheremoisture convergence locally favors convection. As a result subsequent clouds often form in the wake ofprevious clouds. To the extent that the cloud-droplets from a decaying cloud can become incorporated intoa developing cloud, these older, ripened droplets may effectively broaden the droplet spectrum in the newconvective pulse and enhance the initial stages of coalescence growth.

7.2 Endogenous theories of warm rain

7.2.1 Turbulence enhancement to collision efficiencies

Turbulence is thought to modify the collision efficiency of droplets in one of three ways:

1. By modifying the flow-field around the droplets from that which is induced by the droplets themselves.

2. By generating spatial inhomogeneities in the droplet concentration field.

3. By modifying (generally increasing) the relative velocities between droplets.

In considering particle flow interactions an important consideration is the Stokes number, Ns defined asthe ratio of the time-scale it takes for a particle to react to a change in the flow, τp versus the time-scale, tη,for the flow itself to change. So that Ns ≡ τp/tη. For Ns 1 particles react quickly to changes in the flowand follow fluid streamlines. As Ns increases particles increasingly decouple from the flow. The timescaletη is the Kolmogorov timescale, it depends on the intensity of turbulence as measured by the turbulencedissipation rate ε so that

tη =(ν/

η

)1/2

where ν = 1.6×10−5 m s−1 is the kinematic viscosity of the air. Dissipation rates vary by several orders ofmagnitude, but a typical dissipation rate in a cloud may be 1× 10−3 m2 s−3, which yields tη = 0.13s. The

32

particle relaxation time is given as tp = ρlD2/18µg where µg = 1.9 × 10−5 kg m−1 s−1 is the dynamic

viscosity of air, and ρl is the liquid density, so that tp ≈ 3 × 106D2 s. Hence a drop whose diameter is 20µm has an inertial timescale of about 1.2 ms, or a Stokes number of about 0.01. A droplet with a Stokesnumber of unity has the dissipation dependent size of D∗ = 36.5/η1/4 µm, which for typical dissipationrates is about 200 µm.

A priori one would not expect droplets with diameters of 20 µm to be strongly affected by turbulentcirculations, nonetheless numerical calculations which have now become possible at low Reynolds numberdo suggest that the collisional efficiency can be enhanced by all three effects mentioned above, even for therather small Stokes numbers characteristics of cloud droplets in typical clouds, with the net enhancementbeing as much as a factor of 5. However in the critical size range of droplets with diameters of around20-30 µm the enhancement factor is perhaps half of this value, which begins to fall within the uncertainty ofthe collection kernels to begin with. Thus the idea that the modification of the collision efficiency throughturbulence effects is necessary to explain the formation of warm rain remains controversial.

7.2.2 Turbulent mixing

Another idea is that turbulent mixing dilutes the concentration of droplets, thereby reducing the competitionfor the available vapor, and enhancing the growth rate of favored droplets. Turbulent mixing processesare also thought of in terms of competing timescales. If we let tm denote a mixing timescale, and te anevaporation timescale, then if te tm one speaks of the mixing as being inhomogeneous, while if te tmthe mixing is described as inhomogeneous. In both cases the picture is one of mixing between saturatedcloudy air, and unsaturated air. Thermodynamically the mixing of sub-saturated air requires net evaporation,either by reducing the number of droplets in the saturated air after the mixing, or by reducing the size of thedroplets.

dry air

undilute cloudy air

saturatedw/out condensate

saturatedwith condensate

D

dn/d

(logD

)

D

dn/d

(logD

)

final

initial

initial

final

Step 1: Turbulent Stirring Step 2: Mixing followed by evaporation

Hom

ogen

eous

Mix

ing

Inho

mog

eneo

us M

ixin

g

Figure 9: Schematic to show the difference between homogeneous and inhomogeneous mixing. For ho-mogeneous mixing the dry fluid is stirred rapidly through the fluid so that it is effectively homogenizedbefore the droplet spectrum has had a time to react. For inhomogeneous mixing the stirring is slow and onlya portion of the fluid is mixed by the time evaporation has had a chance to thermodynamically adjust themixed fluid.

33

In inhomogeneous mixing, one imagines the dry air saturating through the rapid evaporation of dropletson the edge of an air parcel, before the dry air that is being mixed throughout the parcel has time to mixthroughout the parcel. In the homogeneous mixing limit unsaturated air can be mixed throughout the parcelbefore the droplets have time to adjust. So mixing precedes evaporation and all of the droplets withinthe mixing volume adjust homogeneously to the changing thermodynamic conditions that accompany themixing. In the inhomogeneous limit only a fraction of the droplets feel the intrusion of dry air, and thesereact disproportionately, for instance by complete evaporation. Hence for homogeneous mixing the requiredevaporation is carried disproportionately by the reduction in the average drop size, while for inhomogenousmixing it is carried by a reduction in drop number.

For the case of inhomogeneous mixing, the air in which all the droplets have been evaporated is just sat-urated. This condition follows by definition, as otherwise further mixing would require further evaporation.If subsequent homogenization within the otherwise undilute air is not rapid enough, further upward motionwill result in the activation of additional cloud droplets, thus broadening the droplet spectrum overall. Oth-erwise the condensation occurs on the depleted remaining droplets, thereby reducing the competition for theavailable vapor, and favoring the growth of larger droplets.

8 Lecture 8: Atmospheric ice and its initiation

8.1 The molecular structure of water and ice

Ice crystals depend on the molecular structure of water, which is illustrated in Fig. 10. The two hydrogenatoms bond to the Oxygen atom at an angle of about 105. This structure induces an electric dipole momentwhich is why water vapor is such a strong greenhouse gase. It also constrains the crystalline structure of ice.Solid ice is a crystal, but many different crystalline structures, each in effect defining a different phase ofice are possible. In conditions that typify the troposphere only regular hexagonal ice, denoted Ih, is thoughtto be important. Cubic ice, or Ic, wherein the oxygen atoms arrange themselves in a diamond like structure,can exist at temperatures found in the upper troposphere and at the mesopause, but whether it forms inthe atmosphere remains uncertain. Thus when speaking of ice we take for granted the common form ofhexagonal ice.

0.24 nm

0.28

nm

104.5º

0.095

8 nm

+

+

-

Figure 10: Schematic view of a water molecule.

34

The crystal structure of hexagonal ice is illustrated in Fig. 11. Although shown in a planar view thehexagonal elements consist of alternating up and downward displacements of molecules, which we attemptto illustrate by the alternating colors of the oxygen molecules. Each oxygen molecule has four bonds orga-nized in a tetrahedron like structure, which would imply an angle of 109.5, which is not so different thanthe basic geometry of the H-O-H bonds of the free water molecule. The structure of ice affects its patternsof growth, but the hexagonal shape is manifest for very large crystals. To help describe the growth of icea coordinate system relative to the crystal structure is defined, with the c-axis oriented along hexagonalcolumns, and the a-axes in the basal plane, oriented along the direction of the hexagon defined by oxygenmolecules at the same level (shown by the dashed line in the figure).

lower layer upper layer

a2

a1

a3

1 2

3

45

0.276 nm

Figure 11: Schematic view of hexagonal ice. Shown are the positions and bonds between oxygen molecules(not to scale) in two planar views, where the upper layer lays directly above the lower layer forming hexag-onal columns. Each oxygen molecule has four bonds. The molecules connected to each of these bonds formthe vertices of a tetrahedron. Light colored oxygen molecules are set out of the page with their fourth bonddirected upwards, dark colored molecules are set into the page with their fourth bond directed downward.Hence molecules 4 and 5 bond with one another, and molecules 1, 2, 3 and 4 form the vertices of a tetra-hedron which contains molecule 5 at its center. The a axes of ice align with the ring hexagonal structureshown by the dashed line, the c-axis is oriented out of the page and not shown.

Depending on the local thermodynamic conditions at the surface of the ice-crystal, growth along oneor the other axes of the underlying ice-crystal may be favored. This leads to a wide variety of macroscopiccrystalline forms as illustrated in Fig. 12. Plates naturally arise when the ice-crystal growth along the a-axesis favored, while needles and columns result when c-axes growth is favored. At different stages of the ice-crystal growth thermodynamic conditions may favor one or the other mode of growth, thereby leading tomany apparently hybrid forms, where plates begin growing at the ends of columns to form capped columns,or growth is restricted to the lines of symmetry so that stars may form. Differently shaped ice-crystals willdiffer both in their terminal velocity and in how they scatter radiation. And although no two ice-crystals arethe same, it proves useful to speak of ice-crystal habits as generic classes of ice-crystal shapes that scatterradiation, or have terminal velocities, that are broadly similar. Additional ice-habits are associated withmore amorphous forms of ice, including aggregates, graupel and hail. The number of habits one needs to

35

describe atmospheric ice depends on how demanding ones description may be, but certainly basic habitsinclude: graupel/hail; snow and pristine ice, where the latter is composed of an amalgam of needles, prisms,lates and columns.

8.2 Ice initiation

8.2.1 Homogeneous nucleation

Ice can form homogeneously by either freezing liquid water, or directly through vapor deposition to thecrystalline phase. As is the case for the homogeneous nucleation of liquid water surface tension effectsimply that for a given supersaturation an ice-embryo must reach a critical size. For ice-embryo’s smallerthan this critical size an increase in the crystal size implies an increase in the change of the free energy ofthe system. The change in free energy associated in a change in phase from liquid to solid is given as:

∆G = −N(µl − µi) +Aσli (85)

whereN is the number of molecules,A is the surface area and σli is the surface free energy (surface tension)between the liquid and ice phases, and µ denotes the chemical potential. This expression for the free energyfollows naturally from the definition of µ as the molar specific Gibbs energy and the introduction of a surfaceenergy.

The number of molecules and the area are both related to an effective size, r, of the system, so we canwrite: N = 4/3παr3ni where ni is the number of molecules per unit volume of ice, and α > 1 is a param-eter that accounts for the fact that the ice embryo will not be spherical. Likewise we can define A = 4πβr2.The chemical potentials are defined by kT ln(e) where e is the saturation vapor pressure over liquid (e? orice (ei) as the case may be and k is the Boltzmann constant. Hence, integrating the thermodynamic potentialfrom a reference value at some reference temperature and pressure, T0, p0 isothermally to a pressure char-acteristic of either the ice or liquid phase, e? or ei respectively allows one to write Eq. 85 as a function oftemperature, such that

∆G = −43παr3nikT ln(e?/ei) + 4πr2βσli (86)

Here the temperature dependence is implicit because both ei and e? are functions of temperature only .Defining the critical ice embryo size, r∗ to be that for which ∂(∆G)/∂r = 0, we find that

r∗ =2σli

nikT ln(e?/ei)

(α

β

). (87)

Hence the critical ice-embryo size is a decreasing function of temperature. Colder temperatures requirelarger ice-embryos to nucleate ice, although this is somewhat mitigated by the increase in e?/ei with de-creasing temperature. To calculate the critical embryo size required to form ice by vapor deposition directlyone follows an identical procedure, with the difference being that e? is replaced by the vapor pressure, e andσli is replaced by the surface free energy of the ice-vapor interface, which we denote by σvi.

As is the case for homogeneous nucleation of water drops one can calculate as a function of temperaturethe rate at which embryos of a given size are generated randomly. By associating nucleation of ice with acertain rate of formation of critically sized embryos one arrives at a condition on the temperature and (inthe case of direct vapor deposition) a degree of ice supersaturation. It is found that the temperature forhomogenous freezing of water droplets is 235 K, or about -38 C. In contrast direct depositional freezingonly becomes favorable for temperatures colder than -60 C and at an ice supersaturation ratio (e?/ei) of 15.

36

Figure 12: Sketch of different types of ice crystals taken from SnowCrystals.com.

37

For this reason homogeneous production of ice directly from vapor deposition is not thought to be importantint he atmosphere.

The above considerations must be modified for the fact that super-cooled liquid water is rarely purewater, foremost because it has itself nucleated heterogeneously on a cloud condensation nuclei. Hence,depending on their size, supercooled water droplets in the atmosphere tend to be solutions with differingdegrees of concentration. The effect of the solute is to reduce the water activity. This is what makes theactivation of cloud droplets possible, but it impedes the formation of ice by effectively reducing the waterpressure in the solution drop. However it does make it possible for ice to form in situations where watersaturation has not been achieved, as if temperatures are cold enough small haze particles can still initiatefreezing. For instance even for haze particles with a water activity of 0.9 homogeneous freezing will initiateat temperatures below about -53 C.

8.2.2 Heterogeneous nucleation

Ice is often found at temperatures and ice supersaturations much lower than what is required for homoge-neous nucleation, even for pure water. This leads one to believe that freezing may be facilitated by a catalyst,for instance particulate matter or aerosol particles. However it turns out that to be an effective substrate fornucleating ice the substrate in question must have a molecular structure very close to that of ice itself, andto preserve this structure it must also be insoluble. Particles that allow ice to initiate at temperatures abovethe temperature for homogeneous freezing are called ice nuclei.

e/ei > 1

e/es > 1

Deposition Freezing

Condensation Freezing

Contact Freezing

Immersion Freezing

T < T*i

T < T*t

T < T*d

Figure 13: Schematic showing different modes of ice-nucleation. Modes below the dashed lines are varia-tions on condensation nucleation, distinguishing themselves in terms of how the ice nuclei came into contactwith the super cooled water.

Broadly speaking there are four main mechanisms for heterogeneous nucleation of ice. These are de-picted schematically in Fig. 13. In principle there are two basic mechanism. Deposition freezing directlyon the substrate of the Ice nuclei (IN), or condensational freezing initiated by an ice nuclei. However thedetailed properties of the ice nuclei are important, and how it came in contact with super cooled water mayaffect its ability to act as an ice nuclei. This motivates the distinction between condensation, immersion

38

and contact freezing. In nuclei are more effective if they come into contact with supercooled water withouthaving been previously wetted, that is if we denote by T∗ the activation temperature, then T t∗ 6= T i∗ 6= T d∗ ,and T t∗ > T i∗. Because the ice nucleating ability of a substrate depends so critically on the details of itsstructure and how it may have been modified in the past it is possible to imagine many additional modes ofice nucleation, for instance contact nucleation with ice nuclei that were released through the evaporation ofa previously super-cooled drop may be less effective than contact nucleation involve an ice nuclei that wasnot previously inside of a drop.

Several points can be taken from the above discussion: (i) it is difficult to form ice in the atmosphere,hence it is common to find super-cooled liquid water; (ii) because depositional freezing is relatively ineffi-cient it is more difficult to talk about the ice phase without also considering the liquid phase, than vice versa;(iii) a wide variety of relatively inefficient mechanisms have been identified for identifying ice, and it is notclear if one is dominant since many depend on the detailed structure of a nucleating substrate, or Ice Nuclei.

9 Lecture 9: Ice crystal growth

Ice crystals grow analogously to water drops, that is by collection and diffusion, and the outline of thetheories we developed for these processes are amended, with minor corrections to explain the growth of iceparticles, as described below.

9.1 Diffusional growth theory

Diffusional growth is in principle the same for vapor diffusion to ice particles as it is for water droplets,except for the difference in geometry. This difference in geometry is accounted for by solving the Laplaceequation for the steady state vapor flux in different geometries, to better account for the geometry of thegrowing ice particles. Whereas for the spherical geometry of a droplet, the steady state solution to thediffusion, (35, took the form

n(R) = n∞ − (n∞ − nr)r

R(88)

for other geometries one can write a general solution of the form

n(R) = n∞ − (n∞ − nr)C

R. (89)

which differs through the introduction of the Capacitance, C. Identifying r in Eq. 37 with a capacitance inthe case of solutions for more general geometries stems from the analogy to electrostatics where one solvesfor the electric field about a charged object whose shape is encoded by its capacitance. By approximatingthe ice-crystals as simple shapes the capacitance can be solved for directly, as tabulated in Table 2.

It follows from analogy with the derivation of Eq. 48 that the equation for vapor diffusion to ice can beapproximated as follows

dmdt

= 4πC(S − 1

FD + FK

), (90)

allowing for corrections to FD and FK for kinetic or ventilations effects as are appropriate for ice. Herewe note that in deriving Eq. 90 it is assumed that the vapor flux to the surface of the ice crystal is equalto the vapor flux across a control volume that approximates its shape. This is only approximately true, butappears adequate based on experimental work, although it is noted that where on the ice crystal the vaporis deposited depends on both macroscopic factors that influence the structure of the vapor field, which the

39

Shape Capacitance, CSphere rPlate 2

π r

Oblate Spheroid aε [arcsin ε]−1 where ε2 = 1− b2

a2

Prolate Spheroid A[ln(a+Ab

)]−1where A2 = a2 − b2

Table 2: The capacitance, C for different shaped objects. For the different spheroidal shapes a and b referto the dimension of their major and minor axes respectively, so that A2 and ε2 are greater than 0.

capacitance parameterizes, and microscopic features which depends on how a vapor molecule binds to thegrowing ice surface.

The details of how the vapor gets incorporated into the crystalline structure of the ice are not wellunderstood, and we only give an outline of our understanding below. Clearly this binding depends on localthermodynamic conditions at the surface of the crystal, as is evident from the variety of ice habits one findsin the atmosphere, e.g., Fig. 12. Locally one can differentiate between the two axis of the ice-crystal. Thea-axis lies along the basal plan of the crystal, as shown in Fig. 11, while the c-axis lies perpendicular tothe basal plane, along what is called the prism plane. Under different conditions either a-axis (plate like)or c-axis growth is favored. In the former case we speak of growth on the prism face, versus growth on thebasal face for the latter. Experimental results show that the different mechanisms of growth are favored asa function of temperature, with the rough behavior sketched in Fig. 14. From the figure one would expectcolumns at very cold temperatures, and in the narrow temperature range near -8 C, otherwise plates arefavored.

Temperature [degC]-5 -10 -15 -20

Line

ar G

row

th R

ate

[µm

s-1]

0.2

0.4

0.6

0.8Growth on Basal Face

Growth on Prism Face

Figure 14: Schematic showing growth rates on the basal and prism faces of ice as a function of temperature.

Figure 14 also makes clear that growth is not only along one axis. Although it suggests that the tempera-ture is the only factor that controls the growth of ice, the degree of ice supersaturation is a second factor thathas been used to organize the different modes of Ice growth. In particular as the degree of supersaturationincreases increasingly complex crystals emerge. Because ice often grows in water saturated environmentsthe degree of ice supersaturation is itself often constrained by temperature, however at water supersaturation

40

the vapor will be supersaturated with respect to ice by many tens of percent. As a rule of thumb the icesupesaturation is equal, in percent, to the depression of the temperature below freezing. So from -10 degCto -30C the vapor field just saturated with respect to water increases its supersaturation with respect to iceby from around 10% to 30%. In addition ventilation effects can enhance the effective supersaturation by asmuch as a factor of two. This drives rapid growth and the development of complex crystal structures. Thispoint is shown more clearly in Fig. 15 which attempts to summarize the dominant ice forms as a functionof temperature and ice supersaturation. Arguably the complexity of ice growth is only limited by our abilityto categorize the names. That said, the figure does show the basic pattern of c- versus a-axis growth shownin Fig. 14, for instance with needles and columns evident at temperatures that favor growth along the basalface of the ice.

needle rosettes

longneedles

longneedlesrosettes

long columns

solid columns

solid bullet rosettes

bullet rosettes withlong solid bullets

combinationsof irregulars & columns

short solid columns

assemblages of largeplates an side planes

assemblages of thin plates

plat

es

thick plates

thin plates

sect

or p

late

s

dend

rites

sect

or p

late

s

hollo

w c

olum

ns

need

les

spearheadscolumn plate combinations

crossed plates

gohel twins

side planes

scrolls

effective maximum ice supersaturation due to ventilation effects

water saturation

Temperature [degC]0-10-20-30-40-50-60

Ice

Supe

rsat

urat

ion

[%]

10

20

30

40

50

60

Figure 15: Schematic showing ice growth regimes as a function of ice supersaturation and temperature.This figure is adapted from Fig. 5 of Bailey and Hallet 2009, with the different modes of growth, columnar,plate-like, simple crystals, indicated by green blue and red labeling respectively.

9.2 Ice crystal terminal fall speeds

The complexity of ice habits also complicates the calculation of ice-crystal fall speeds. While for ideal-ized crystal shapes one could imagine computing or measuring ice fall speeds, the variability in ice habitis sufficiently diverse to discourage such an enterprise. Hence ice-crystal fall speeds tend to be crudelyparameterized as a function of habit and effective diameter according to the following formula

u∞(D) = αDβ (91)

where α and β depend on the habit, and D is the diameter of a drop with the same effective water content.This choice of D, likely reflects the measurement technique. As a rule β < 1, with typical values rangingfrom 0.2 to 0.6, with the larger exponent corresponding to Graupel. For many crystal shapes β ≈ 1/3 isa good approximation, which emphasizes that crystal fall speeds do not vary strongly as a function of size,and hence the prefactor α plays a more important role. Overall ice crystals larger than 1mm in effectivediameter have fall speeds that vary between 0.5 and 2.0 ms−1, with a typical fall-speed for a snowflakearound 1 ms−1.

41

9.3 Growth by collection

Ice also grows by gravitational collection. However collection growth for ice differs from that for waterdroplets in that for ice, diffusional growth theory is sufficient to explain the presence of collection embryos,those that by falling through a field of smaller droplets or ice-crystals, continuously collect them. Hencethe collection growth of ice is usually based on the continuous collection equation as opposed to the quasi-stochastic, or Smoluchowski theory. This equation posits that the collection of mass by a collector embryoevolves as

dMdt

=Eπ

4

(π6ρindd3

)(D + d)2(u∞(D)− u∞(d)) (92)

which follows in direct analogy to the theory developed for droplets. Here M denotes the collector dropmass, while π

6ρindd3 is the mass concentration of the collected drops, with nd denoting the population

density and the overline implying an average over the distribution. Here the dimension of the collectorand collected particle is distinguished by upper and lower case. In most cases of interest the fall velocities,collection and coalescence efficiencies are poorly known, so that the approximation that (D2+d2)(u∞(D)−u∞d) ≈ D2u∞(D) is a good one, which further simplifies the formula.

Because ice is difficult to form, mixtures of ice and liquid water are common. As a result it provesuseful to distinguish growth by collection by what is being collected. Riming refers to the process wherebyice-crystals collect super-cooled water drops which then freeze when they come into contact with ice. Aggre-gation refers to snowflakes collecting one another and clumping together. Note that the irregular motion ofsnowflakes increases the aggregation efficiency among like-sized particles, this also motivates a distinct termfor this process. Lastly accretion refers more generally to larger ice crystals collecting smaller ice-crystals.

9.4 Further ice-microphysical processes

The variety of ice habits, its very non-equilibrium behavior, and its tendency to exist with liquid water inmixed-phase environments over a wide range of temperatures leads to a variety of fascinating ice microphys-ical processes. These range from ice multiplication associated with the splintering of ice-crystals during theriming process within a certain temperature range. Ice multiplication can greatly confuse attempts to re-late ice-concentrations to ice nuclei measurements, particularly for relatively warm clouds where it is mostfavored.

Another interesting aspect of ice is the so-called seeder feeder mechanism, whereby small orographicwater clouds, that themselves do not precipitate, can greatly enhance precipitation rates from overlayingprecipitating ice clouds. As the ice falls through even a small water cloud it grows rapidly in the watersupersaturated environment. Such a process can enhance rain rates by a factor of two, and is common inorographic environments as the water cloud often is forced by even small changes in topography.

10 Lecture 10: The atmospheric aerosol

The atmospheric aerosol is the collection of non-cloud, supended particulate matter in the atmosphere.Strictly speaking an aerosol is nothing more than the suspension of one phase of matter within another –typically liquid or solid particulate matter suspended in a gas. By this strict definition, water and ice cloudsare also an aerosol. However because it is interesting to understand the properties of particulate suspensionsoutside of clouds, the term aerosol, or aerosol particles, is reserved for suspensions of non-cloud particulatematter in the atmosphere.

42

Size Junge’s Categorization Whitby’s Categorizationr ≤ 0.1 µm Aitken Particles Nucleation Mode Particles0.1µm < r ≤ 1 µm Large Particles Accumulation Mode Particles1µm < r Giant Particles Coarse Mode Particles

Table 3: Size based aerosol categorizations that have been proposed in the literature. In the first column thesize is measured in terms of the dry radius r.

The atmosphere accomodates an enormous diversity of aerosol particles, the classification of which isstill incomplete. Quite early it was appreciated that the composition of the aerosol over land differed fromthat over ocean, and that over land the aerosol in turn varied according to the land surface properties andprocesses. But over the years, continuing through the present, the aerosol has been much more systemati-cally categorized, in terms of one of a number of qualities. For instance, the aerosol can be categorized bysize, by source, by composition or by function. Early categorizations were size based, for instance two sizebased categorizations are presented in Table 3. Junge’s classification scheme is used interchangeably withthe more process based categorization scheme introduced by Whitby. With the hypothesis that very largeaerosol particles may be important for rain formation in warm clouds, the term ultra-giant aerosol particleshas also been introduced to distinguish particles with dry diameters larger than 10 µm. Additionally, in theWhitby categorization scheme one speaks of both accumulation and nucleation mode aerosol as belong-ing to the fine mode. Junge named his smallest category of aerosol particles for John Aitken, a scottishmeteorologist who introduced the concept of cloud condensation nuclei near the end of the 19th century.

The Junge distribution suggests that aerosol mass is roughly distributed equally across size ranges, sothat the mass of Aitken particles is roughly the same as the mass of large, or giant aerosol particles. Thiscan vary from place to place, but is a good rule of thumb, and suggests that the concentration is domiantedby the smallest particles.

10.1 Types of aerosol particles

Aerosol particles can also be distinguished by the way in which they are produced. Broadly speaking twomechanisms are identified. Primary and secondary production. Primary production refers to mechanismswhich mobilize particulate matter directly, for instance from the bulk. Examples would be through thedisintegration of the land-surface, or from the decomposition of biomass, or from sea-spray. Secondaryproduction refers to the formation of particles from the gas phase, through condensable vapors. Aerosolnucleation, or the secondary production of the aerosol, is responsible for producing small particles, hencethe concept of the nucleation mode. Primary production of aerosol can produce particles of all size ranges,as lofted dust and sea-salt particles can classify as giant or coarse mode aerosol.

As more advanced instrumentation has been developed more has been learned about aerosol composi-tion, and broad compositional categories have developed, as discussed below.

Sea Salt: Sea-salt, NaCl, is very hygroscopic an thus acts effectively as a cloud condensation nuclei. Thesea-salt aerosol is formed by the bursting of spray droplets at the ocean surface. The spray droplets that aremost effective at producing long-lived sea-salt aerosol particles are those that form when bubbles burst onthe surface of the ocean. Hence the roughening of the sea is important for producing sea-salt aerosol not

43

because it mechanically breaks the water surface into a fine spray that evaporates, for instance in associationwith breaking waves, but rather because these breaking waves force air into the water, which then nucleatesbubbles that move to the surface and break. The fine spray from these breaking air-bubbles is believe to bethe source of most of the sea-salt aerosol.

Mineral Dust: Mineral dust generally refers to silicate particles mobilized from the land surface. Dustparticles tend to be large, or giant, and because they are often insoluble, may function as effective ice nuclei.Sources of mineral dust are usually deserts and semi-arid regions, with areas in North China or Mongolia,and North Africa being well known. Dried lake beds are thought to be an important source of fine sedimentthat can be readily mobilized into the atmosphere. Due to a favorable combination of meteorology andsurface properties, one such area, the Bodele Depression on the edge of lake Chad, is thought to supply alarge fraction of the dust exported from the Saharan desert.

Sulfate: Broadly speaking a sulfate is a salt of sulfuric acid. These include sulfuric acid H2SO4, and Am-monium sulfate, or (NH4)2SO4, both of which are inorganic salts, and dimethylsufate (DMS), (CH3O)4SO2,which is an organic salt. Sulfate aerosols tend to be very hygroscopic and thus are very effective cloud con-densation nuclei. They arise from a variety of sources ranging from combustion processes to volcanism to,in the case of DMS, the decay of phytoplankton in the ocean. The burning of fossile fuels releases sul-fur dioxide which then produces sulfate aerosols, hence a significant component of the sulfate aerosol isassociated with anthropogenic activity.

Black Carbon: Black carbon is produced from combustion processes, in particular incomplete natural oranthropogenic biomass burning, as well as industrial processes. Although it is estimated that the mass ofblack carbon in the atmosphere is anywhere from one to ten percent of the mass of sea-salt, it has comeunder increasing attention because of its potential role as an absorber of atmospheric radiation. To theextent that sources of black carbon are related to human activities it, like the sulfate aerosol, may contributesignificantly to anthropogenic climate change.

Organic Aerosol: The organic aerosol is a loose term for a rich variety of organic matter that is foundin the matter. Usually it is distinguished from black carbon on the basis of its light absorption/scatteringproperties. Recent work suggests that organic aerosol particles, or organic coatings of other aerosols, maybe responsible for more than 50 % of the total aerosol mass over continental areas. Much of the organicaerosol is thought to be the result of condensation of the gas phase, with organic gases being produced bythe terrestrial biosphere, although the bulk production of organic particulate matter from decaying plants,or through the emission of pollen may also be important. Hence the organic aerosol, of which DMS is acomponent of, is thought to be a potential link between the biosphere and atmospheric processes and an areaof great current interest.

10.2 Abundances

The primary production of aerosol particles is essentially limited to surface processes, hence it is not sur-prising that the bulk of the aerosol mass is concentrated in the atmospheric boundary layer. This lowest,roughly 1 km of the atmosphere, contains about 10 % of the atmospheric mass, but as much as 80 % of theaerosol mass. Assuming a Junge distribution about one third of the mass of the aerosol is contained in the

44

coarse mode, and two thirds of the mass in the fine mode. The coarse mode, with its greater settling velocityis more concentrated close to its sources at the surface.

Figure 16: Aerosol measurements compiled by Clarke and Kaputsin (Science, 329, page 1488, 2010). Datarepresent aircraft measurements over extensive areas, with each point representing roughly 1 km of data.The points are colored by CO, which is a pollution proxy, and the scattering cross section for visible light,σsc, in the second columns of panels is scaled by the angstrom exponent, α. Large values of the angstromexponent are indicative of small fine-mode aerosol associated with combustion. CNHot referes to particleconcentrations measured after being heated to 300 C which removes the volatile aerosol, and is argued bythe authors to be a good proxy for the CCN.

Measured by concentration fine mode aerosol are also more pronounced near the surface, but lessmarkedly so. Measurements suggest that between 5 and 10 km the distribution of Aitken, or nucleationmodel, particles is rather uniform. Junge discovered that large, or accumulation mode, aerosol particles fol-low the profile of the smaller Aitken particles through the troposphere, but have a concentration maximumin the lower stratosphere. In a layer between 15 - 25 km accumulation mode particles show a maximumconcentration in what has come to be called the Junge layer, and wherein aerosol concentrations are on theorder of 100 L−1, some five to ten times larger than their background concentration in the upper troposphere.Given a rule of thumb that the mass of accumulation and nucleation mode particles are the same, this impliesthat background concentrations of Aitken particles range for 10-100 cm−3 in the free troposphere.

Typical concentrations of aerosol particles over the ocean are around 300 cm−3, but vary considerably,by as much as a factor of three. Only a fraction of these serve as CCN, and so as rule of thumb maritimecloud-drop concentrations are around 100 cm−3. Over the continent concentrations are much higher, withconcentrations of around 10000 cm−3 near urban areas, approaching ten times this much in very pollutedregions. Many of the observed properties of atmospheric aerosol particle abundance are summarized withthe help of Fig. 16.

45

10.3 Aerosol Processes

Aerosol processes are similar to those previously discussed for the case of cloud droplets. Aerosol particlescan be created directly through primary production mechanisms, or nucleated from the gas phases. Aerosolparticles grow by vapor diffusion, particularly organic aerosols where large sources of gas-phase organicsmay be present in regions of industrial pollution. Aerosol particles can also grow as a result of cloudprocessing. Droplets forming on an aerosol particle will increase the total amount of solute they cary bysubsequently collecting other droplets, or through the uptake of soluble gases, leading to the increase ofthe total amount of solute in the gas. Upon evaporation a cloud droplet has been found to leave behind asingle aerosol, which will be accordingly larger than the aerosol on which the original droplet nucleated.Moreover, the resultant aerosol may be considerably more complex in terms of composition and shape thanthe original one. The primary mechanism of aerosol growth is, however, coagulation.

The nucleation mode aerosol is not efficiently collected by cloud droplets, because their lack of inertiaenables them to move with the flow around droplets that fall toward them. However this lack of inertia meansthat their motion is easily influenced by random bombardments by gas molecules, thus endowing the aerosolwith Brownian motion that leads to collisions and coagulation. The formalism for describing coagulationgrowth of the aerosol is identical to that for the quasi-stochastic collection among cloud drops, except thatthe interaction kernel is not the gravitational kernel derived for cloud drops, rather the coagulation kernel:

K(x, y) = 4π(Dx +Dy)(rx + ry). (93)

The diffusion coefficient, D for a particle of radius r is D = kT/(6πηr) where η is the dynamic viscosity.Substituting this expression into the collection kernel yields

K(x, y) =2kT3η

(1rx

+1ry

)(rx + ry) ≈

2kT3η

(ryrx

)(94)

for rx ry. Thus showing how small particles can be effectively collected by larger ones. However smallparticles may coagulate on cloud droplets as they are pushed into the drop as a result of molecular collisions.Coagulation is the primary sink of nucleation mode particles, and an important source of accumulation modeaerosol particles.

Coagulation is not a sink or source of aerosol mass. The aerosol mass budget is balanced by deposition,which can either be wet, or dry. Dry deposition occurs when aerosol particles grow large enough to have ap-preciable fall velocities, or by coming into contact with a surface through brownian motion. Wet depositionrefers to the deposition of aerosol mass that is dissolved in hydrometeors that fall to the ground. The largedifferences between the continental and maritime aerosol has as much to dow with aerosol sources beingmore prevalent over land, as it does with aerosol sinks being more prevalent over the ocean. The co-varianceof aerosol sources and precipitation processes is often overlooked, but is crucial to properly predict aerosolconcentrations. While it is tempting to interpret the lack of aerosol in regions of precipitation as a cause ofthe precipitation, through the effects of a depleted aerosol on cloud microphysical processes, however theopposite interpretation is usually more correct. Precipitation, which may form simply because for dynamicreasons so much moist air is forced upward that it must rain, is the reason the aerosol is depleted.

10.4 Aerosol Function

In addition to size, composition, or source type, aerosol particles can also be described in terms of function.Broadly speaking four different types of functions have been identified for the aerosol. Aerosol particles

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may be cloud active, in which case they serve as cloud condensation, or ice, nuclei. Aerosol particles mayalso be radiatively active, in that they efficiently scatter or absorb sunlight. Although aerosol particles alsointeract with terrestrial radiation usually their interactions with solar radiation are emphasized. An examplebeing in the case of Volcanoes, where the production of sunlight from the huge injection of sulfur, andsubsequent production of sulfate aerosol, can remain in the lower stratosphere for more than a year andsignificantly depress global mean temperatures. Interestingly, for very large volcanoes the long-wave effectof the aerosol increasingly offsets its shortwave effect with the net result being that the aerosol impact fromVolcanos scales sub-linearly with the size of the volcano.

11 Lecture 11: Microphysical models of cloud and aerosol

In this lecture we will not endeavor to give a complete description of clouds and aerosol models, butrather outline different approaches, and their motivation. Broadly speaking one can identify three typesof cloud/aerosol models:

• particle based methods;

• non-parametric distribution methods;

• parametric distribution methods.

Each of these are discussed in turn below.

11.1 Particle based methods

Models based on particle methods treat each aerosol individually, and evolve a dynamic equation describingessential qualities of each particle. The simplest example of particle model would be that which describesone aerosol particle with a fixed amount of soluble mass, for instance sea-salt. For such a description theonly degree of freedom is the amount of water mass is associated with the particle. However, given that thedensity of liquid water is known it is sufficient simply to write an equation for aerosol position and size, thelatter follows directly from Eq. 48, and assumes that the supersaturation is known as a function of space (x)and time:

dx(r, t)dt

= v (95)

drdt

=1r

(S − 1

FD + FK

). (96)

To extend this description to a population of N particles involves simply increasing the number of equa-tions, so that xi,vi describes the position and velocity of the ith particle and ri is its radius, where i =1, 2, . . . N. This type of description can naturally be extended to include a description of other parti-cle attributes, like chemical composition, thereby allowing a consideration of chemical processes that maychange the dry aerosol mass properties. Additionally it may be desirable to allow the growth of the particlesto interact with the vapor field, so that instead of prescribing a supersaturation this is predicted by assumingthat all the aerosol particles are contained within an air parcel in local equilibrium so that its thermodynamicproperties can be described by its temperature, moisture content and density. Such models are called parcelmodels and are frequently used to study the activation of aerosol particles, as they allow for the competitionfor vapor among the N particles in a parcel of some fixed mass of air.

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One limit of particle based methods is the treatment of processes which change the number of particles,for instance coalescence or the production of aerosol mass, as these require the system of equations to bedynamic. This in principle is only a small complication, the real limitation of such approaches is that thenumber of particles that must be solved for is exceedingly large. Assuming aerosol concentrations on theorder of 1000 cm−3 simply solving for 1 m−3 of air implies the need to solve 2 × 109 equations. Hence touse such methods to represent even the smallest of clouds is inconceivable without further approximation.

11.2 Distribution based methods

To overcome the shortcomings of particle models one can take advantage of the great number of particles byassuming that they are continuously distributed over their physical parameter of interest, for instance theirsize. Based on this assumption one can solve for the evolution of the distribution function of the particles,i.e.,, n(x) the number concentration as a function of particle mass, x, rather than the particles themselves.These methods require one to solve for the evolution of the distribution function, namely

∂n

∂t+∇ · (v + u∞(x)n) +

∂

∂x

(dxdtn

)=

12

∫ x

0n(x− y, t)n(y, t)K(x− y, y) dy

−n(x, t)∫ ∞

0n(y, t)K(x, y) dy,+A, (97)

where we have introduced coalescence as one of the source terms in Eq 71, as well as an additional sourceterm A chosen to represent the activation, or deactivation, of particles. Further note the distinction betweenx and y which are used to denote mass, and x which was previously used to denote particle position. In thisdescriptions particles are assumed to move with a velocity that is the sum of the fluid velocity, v, and theterminal fall velocity of the particle. One advantage of such methods is that particle sources and sinks canbe easily incorporated without changing the degrees of freedom in the representation.

A disadvantage of Eq. 97 is that in practice it is difficult to generalize for changes in the particles otherthan a change in mass. If one is interested in tracking both particle mass and solute amount, then a twodimensional distribution function over water mass and solute mass must be solved. And if more than onetype of solute is allowed, or if particles change their shape as might be the case for ice clouds, then one mustin general solve for a multidimensional distribution function. This becomes computationally very intensive,and the assumption that the particles are continuously distributed over solute type and particle shape mayalso become questionable. Even without these complications Eq. 97 can be challenging to solve because itinvolves discretizing the distribution function n(x) over a very large range of particle sizes.

11.3 Parametric distributions

To avoid some of the difficulties of full distribution based methods one often assumes at the outset a cer-tain distribution shape. Then instead of discretizing the distribution itself, the distribution can be solvedgiven knowledge of its parameters, and their evolution. For instance, if one assumes that cloud-dropletsare distributed log-normally, then the entire distribution can be reconstructed from knowledge of just threeparameters: the mean size, the standard deviation of sizes, and the total number of particles. These param-eters in turn can be related to moments of the distribution. This reduces the problem to solving a relativelysmall number of equations: one for the evolution of particle number, another for evolution of mean par-ticle surface area, and a third for evolution of mean particle mass, but comes at the cost of assuming thatthe particles are always distributed in a certain manner. Although observed particle distributions can oftenbe quite well represented by log-normal or gamma distributions, it is often the small deviations from such

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distributions which are key to the evolution of the distribution, for instance by initiating coalescence. Moreaccuracy can be obtained by using more complex distribution functions, for instance the composition ofseveral log-normal distributions confined to different size ranges, or associated with different habits, butmore complexity comes at the cost of additional parameters that must be solved for.

Despite their limitations, assumed distribution methods are usually the only microphysical descriptionwhich lend themselves to incorporation into dynamic models capable of representing cloud-scale circula-tions. In such models it is common to define habits for cloud water and cloud ice, precipitation, snow andgraupel, although depending on interest other particle habits, such as hail and intermediate forms of ice,may also be introduced. Simplifications of the distributions associated with each habit are also common, forinstance by fixing the width of the distribution, or using simpler one parameter distribution functions, onereduces the degrees of freedom in the problem, albeit at the expense of some accuracy.

12 Lecture 12: Aerosol and cloud measurement systems

12.1 In situ Methods

12.1.1 Single particle sampling

12.1.2 Expansion Chambers

12.1.3 Particle Counters

12.1.4 Particle Spectrometers

12.2 Active remote sensing

12.2.1 Cloud and Precipitation Radars

12.2.2 Lidars

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