the inï¬‚uence of radiation on ice crystal spectrum in the

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 134: 609–620 (2008)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.226

The influence of radiation on ice crystal spectrum in theupper troposphere

Xiping Zeng*Goddard Earth Sciences and Technology Center, University of Maryland, Baltimore County, and Laboratory for Atmospheres, NASA Goddard

Space Flight Center, Greenbelt, Maryland, USA

ABSTRACT: This theoretical study is carried out to investigate the effect of radiation on ice crystal spectrum in the uppertroposphere. First, an explicit expression is obtained for the ice crystal growth rate that takes account of radiative andkinetic effects. Second, the expression is used to quantitatively analyse how radiation broadens the ice crystal spectrumand then to reveal a new precipitation mechanism in the upper troposphere and the stratosphere. Third, the radiative effectis used to explain the subvisual clouds near the tropopause. Copyright 2008 Royal Meteorological Society

KEY WORDS cirrus; subvisual cloud; precipitation

Received 18 March 2007; Revised 19 January 2008; Accepted 23 January 2008

1. Introduction

Tropical precipitation is still a challenge in meteorology(Simpson et al., 1988). It involves multi-scale processes.Small-scale precipitating convection has a propensity tospontaneously organize into coherent structures or cloudclusters (e.g. Ludlam, 1980; Moncrieff, 2004), and cloudclusters in turn generate cloud shields in their matureand later stages (Zipser, 1969; Houze, 1977, 1982).Since cloud shields are widespread (or 104 –105 km2 inarea; Houze, 1982; Nesbitt et al., 2000), they modulateatmospheric radiation and thus change large-scale verticalcirculations (Houze, 1982). As a feedback, large-scalecirculations modify clouds and precipitation (Raymondand Zeng, 2005). In this multi-scale feedback, convectiveclouds and large-scale circulations are highly coupledthrough upper-tropospheric cirrus (e.g. Raymond andZeng, 2000).

Theoretically speaking, cloud-resolving models can beused to evaluate the cirrus in the coupling betweenconvective clouds and large-scale circulations. However,recent evaluations of cloud-resolving model simulationsshowed that the models predicted excessive clouds espe-cially in the upper troposphere (e.g. Petch and Gray,2001; Zeng et al., 2007). Moreover, Starr and Quante(2002) compared the results of cirrus models using binand bulk representation of microphysical processes. Theypointed out that the bulk-parametrization model diag-nosed a significant population of large ice particles whilethe bin model was dominated by small ice particles.Those modelling studies suggest that some physical pro-cesses are overlooked in the current cloud models.

* Correspondence to: Xiping Zeng, Mail Code 613.1, NASA/GoddardSpace Flight Center, Greenbelt, MD 20771, USA.E-mail: [email protected]

Recent aircraft observations of ice particles (e.g.Heymsfield, 1986; Heymsfield et al., 2006) have pro-vided clues as to the processes that were overlooked inthe current cloud models. Heymsfield (1986) showed thatlong-lasting subvisual clouds near the tropical tropopauseconsisted of ice crystals with small size and low num-ber density. Heymsfield et al. (2006) reported that themaximum measured ice particle diameter increased withair temperature from −70 to −85 °C when convectionexisted below but decreased in the absence of deep con-vection.

Those modelling and observational issues may involvecloud–radiation interaction (see section 6 for discussion).Previous studies have proposed that radiation affectsthe diffusional growth of ice crystals (Hall and Prup-pacher, 1976; Stephens, 1983; Hallett, 1987; Hallettet al., 2002). Stephens (1983) showed that radiation sig-nificantly increases the survival distance of falling icecrystals in an unsaturated environment. He further sug-gested that the radiative effect on ice crystal growth beintroduced in cloud models. However, few current cloudmodels (Wu et al., 2000) account for it. One possible rea-son is that the ice crystal growth rate with the effect isexpressed implicitly through the ice crystal surface tem-perature, making it less easy for a cloud model to adopt.Hence, it is imperative to obtain an explicit expressionfor the ice crystal growth rate first.

Accordingly, the radiative effect is studied in thepresent paper beginning with an explicit expression forthe ice crystal growth rate. The expression is thenused to quantitatively analyse how radiation broadensthe ice crystal spectrum and finally to propose a newprecipitation mechanism in the upper troposphere. Thepaper consists of seven sections. Section 2 describes atheoretical framework for ice crystal growth that takes

Copyright 2008 Royal Meteorological Society

610 X. ZENG

account of radiative and kinetic effects. Sections 3and 4 introduce linear and nonlinear models to showhow radiation affects ice crystal spectrum, respectively.Section 5 presents a condition for haze aloft. Section 6uses the present results to explain some clouds in theupper troposphere. Section 7 concludes. Except whenspecified, the paper follows the symbol definition inappendix A.

2. Theoretical framework

2.1. Heat balance of an ice crystal

The heat balance of an ice crystal in a rarefied atmo-sphere is represented by the following form (Hall andPruppacher, 1976; Roach, 1976; Stephens, 1983; Hallett,1987; Hallett et al., 2002):

Ls4πCDFβ{ρsi(Ts) − ρv}fm = 4πCKFα(Te

− Ts)fQ + SFs − Sε0σ(T 4s − εrηT 4

e ), (1)

where Ts and Te refer to the ice crystal surface andthe environmental air temperature, respectively; ρv andρsi denote the density of the ambient water vapour andsaturated water vapour near a crystal, respectively; S

denotes crystal surface area; fm and fQ are the ventilationfactors expressed in Hall and Pruppacher (1976). Otherfactors are:

(1) C – the stationary diffusion shape factor thataccounts for the influence of crystal shape on thewater vapour field around a crystal. For a sphericalcrystal of radius r , C = r .

(2) D – the diffusivity of water vapour in air that isdescribed by Hall and Pruppacher (1976):

D = 2.11 × 10−5(T /T0)1.94(p0/p) [m2 s−1].

(2)

(3) K – the coefficient of thermal conductivity of dryair that is calculated with the formula of Beard andPruppacher (1971)

K = {2.38+0.00703(T −T0)}×10−2[J m−1 s−1 K−1],

which is independent of air pressure even when air pres-sure is low (N. Fukuta, 2004, personal communication).(4) Fβ – a factor that accounts for the kinetic effect on

water vapour diffusion. It is expressed as (Hall andPruppacher, 1976)

Fβ = r∗/(r∗ + l∗m) (3)

where

r∗ = S/4πC (4)

l∗m =(

2π

RvTs

)1/2Dfm

2β(2 − β)−1 . (5)

(5) Fα – a factor that accounts for kinetic effect onheat diffusion. It takes the form (Fukuta and Walter,1970; Carstens, 1972; Carstens et al., 1974; Hall andPruppacher, 1976; Pruppacher and Klett, 1997)

Fα = r∗/(r∗ + l∗Q), (6)

where

l∗Q =(

2π

RdTs

)1/2KfQ

ρaCp2α(2 − α)−1 . (7)

The factors 2β(2 − β)−1 and 2α(2 − α)−1 in (5) and(7) are referred to as the apparent deposition and thermalaccommodation coefficients, respectively, (Chapman andCowling, 1970; Fukuta, 2004 (In a copy of his manuscript‘Advanced Cloud Physics’ received in 2004, N. Fukutaused the representative deposition coefficient and thermalaccommodation coefficient to name the two factors,respectively.)) Based on experimental measurements (e.g.Fukuta and Walter, 1970; Shaw and Lamb, 1999), thethermal accommodation coefficient α = 0.7, and thedeposition coefficient β = 0.04 exp{−(T − T0)/85} thatfits the data of Fukuta and Walter (1970) when T >

−85 °C.(6) Fs – the solar flux absorbed by an ice crystal.(7) ε0 – the bulk absorption efficiency of an ice crys-

tal for blackbody radiation. It is defined as the ratioof the infrared flux absorbed by a crystal to theradiative flux emitted by a blackbody at the crys-tal surface temperature. Based on the results ofStephens (1983), the efficiency is approximated withε0 = 0.9{1 − exp(−r∗/6)} for a crystal surface tem-perature of Tref = 261 K, where r∗ is measured inmicrometres. The Wien displacement law shows thatthe wavelength of peak emission for a blackbody isproportional to the reciprocal of temperature. Sinceradiation absorption depends on the dimensionlesssize parameter 2πr∗/λ where λ is radiation wave-length, the bulk absorption efficiency at temperatureT is approximated with

ε0(r∗, T ) = 0.9{1 − exp(−T r∗/6Tref)}. (8)

(8) εr – the relative bulk absorption efficiency of an icecrystal for infrared radiation. It is defined as ε/ε0

where ε represents the ratio of the infrared fluxabsorbed by a crystal to the infrared flux incident onthe crystal. Let Tmin and Tmax denote the minimumand maximum temperature in the atmosphere andits underlying surface. The radiation emitted by ablackbody at temperature Tmin and Tmax may impingeon a crystal as two extreme limits. Hence,

ε0(r∗, Tmin)

ε0(r∗, T )

< εr(r∗, T ) <

ε0(r∗, Tmax)

ε0(r∗, T )

,

which becomes

Tmin/T < εr < Tmax/T

Copyright 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 134: 609–620 (2008)DOI: 10.1002/qj

RADIATION AND ICE CRYSTAL SPECTRUM 611

with the aid of (8). Obviously, εr varies around unity.For the sake of simplicity, εr = 1 is assumed except insection 5.(9) η – the ratio of the infrared flux incident on crystal

surface to the radiative flux emitted by a blackbody atthe environmental air temperature. It is approximatedwith

η = F + + F −

2σT 4e

, (9)

where F + and F − represent the upward and downwardfluxes of infrared radiation, respectively. Given a verticaldistribution of atmospheric variables, the fluxes and thusη are determined by radiative processes (e.g. Fu and Liou,1992; Chou et al., 1995).

2.2. An explicit expression for ice crystal growth rate

The growth rate of an ice crystal is expressed in termsof ice crystal surface temperature Ts (e.g. Hall andPruppacher, 1976). That is,

dm

dt= 4πCDFβ{ρv − ρsi(Ts)}fm, (10)

where m is crystal mass. Next, (1) is used to eliminateTs in (10) to derive an explicit expression for dm/dt .

To simplify the explicit expression and its derivation,a critical case of an ice crystal in a specific environmentis considered first. In the environment, air temperature isstill Te, but humidity is so specific that the crystal doesnot grow or shrink. Let the symbol Tsc denote the crystalsurface temperature, and ec and Hic denote the partialwater vapour pressure and relative humidity in the criticalcase respectively, (The critical relative humidity Hic isdifferent from the effective radiative relative humiditydefined by Hall and Pruppacher (1976). The latter isassociated not only with the radiative effect on ice crystalgrowth but also the diffusivity of water vapour in air.).Thus, the crystal in the critical case has a heat balancebetween radiative cooling and conductive heating. Thatis,

Sε0σ(T 4sc − εrηT 4

e ) − SFs = 4πCKFα(Te − Tsc)fQ.

(11)

Since water vapour is spatially uniform in the criticalcase,

ρsi(Tsc) = ec(RvTe)−1. (12)

After introducing the symbol

γ ≡ (Te − Tsc)/Te (13)

and the approximation

T 4sc ≈ T 4

e + 4T 3e (Tsc − Te), (14)

(11) is solved with

γ = χ

4(1 + χ)

(1 − εrη − Fs

ε0σT 4e

)(15)

where the symbol χ , as introduced in (20), represents theratio of the infrared flux to the conductive heat flux fromair to an ice crystal, where both air and ice crystal aretreated as blackbodies.

χ = Sε0σT 3e

πCKFαfQ

. (16)

Using (12) and the ideal gas law for water vapour, thecritical relative humidity Hic = ec/Esi(Te) is rewritten as

Hic = TeEsi(Tsc)

TscEsi(Te). (17)

This expression is changed further to

Hic = 1

1 − γ

(1 − Lsγ

RvTe

)(18)

with the aid of (13), (15) and the approximation

Esi(T )

Esi(Te)≈ Ls

Rv

T − Te

T 2e

+ 1 (19)

that is obtained from the Clausius–Clapeyron equationfor when T is close to Te.

Starting with expression (18) for the critical case, anexplicit expression is derived next for the growth rate ofan ice crystal in a general environment. Subtracting (11)from (1) yields

Ls4πCDFβ

{ρsi(Ts) − (eTsc/Te)

RvTsc

}fm

= 4πCK(1 + χ)Fα(Tsc − Ts)fQ, (20)

using (14), (16) and the ideal gas law for water vapour.Equation (20) is still equivalent to (1) because the knownvariable Tsc always satisfies (11).

Fortunately, (20) is formally equivalent to the energyequation of the same crystal in an environment that ischaracterized as having an air temperature Tsc, partialwater vapour pressure (eTsc/Te), coefficient of thermalconductivity K(1 + χ), and no radiative effect. Derivingthe explicit expression for ice crystal growth rate, or theleft side of (20), just as done in many textbooks on cloudphysics (e.g. Rogers and Yau, 1989) gives

dm

dt= 4πC

A(Tsc) + B(Tsc)

{(eTsc/Te)

Esi(Tsc)− 1

}, (21)

where

A(T ) = Ls

FαfQK(1 + χ)T

(Ls

RvT− 1

)(22)

B(T ) = RvT

FβfmDEsi(T ). (23)


612 X. ZENG

To remove Tsc from (21), A(Tsc) and B(Tsc) are changedto

A(Tsc) ≈ A(Te)T 2

e

T 2sc

B(Tsc) ≈ B(Te)TeEsi(Te)

TscEsi(Tsc)

with the aid of (2) and |Tsc − Te| � Te. Substituting (17)and the preceding two approximations into (21) yields anexplicit expression for the growth rate of an ice crystalin a general environment. That is,

dm

dt= 4πC(Hi − Hic)

(1 − γ )2{HicA(Te) + B(Te)}, (24)

which does not involve Ts. Expression (24) shows that anice crystal grows when relative humidity Hi > Hic andshrinks when Hi < Hic. If εrη = 1 and Fs = 0, Hic =100%, which reduces the expression into the traditionalone without the radiative effect (e.g. Rogers and Yau,1989).

Infrared and solar radiation affect Hic. Due to γ � 1and χ � 1, (15) and (18) show that

(1 − Hic) ∝(

1 − εrη − Fs

ε0σT 4e

).

Since infrared absorption of ice crystals is more importantthan solar absorption, the effect of solar radiation on icecrystal growth is neglected in the paper (i.e. Fs/ε0σT 4

e =0). Next, Hic is discussed by dividing (1 − Hic) into twofactors: (1 − εrη) and (1 − Hic)/(1 − εrη). The formerfactor involves radiative processes and the latter onedepends on crystal size, shape and other non-radiativevariables.

2.3. Infrared ratio η

The infrared ratio η approximately measures the relativemagnitude of inward and outward infrared fluxes on thesurface of an ice crystal. When εrη < 1, Hic < 100% andthus radiation enhances crystal growth. Otherwise, Hic >

100% and radiation benefits crystal shrinkage. Since η

varies with the variables air temperature, water vapourand clouds, the following example is presented to showhow η varies with height through the variables.

Consider the tropical atmosphere observed during theKwajalein Experiment (KWAJEX) that was conductedaround Kwajalein Atoll from 23 July to 15 Septem-ber 1999. Figure 1 shows the mean air temperature andmixing ratio of water vapour versus pressure. Air tem-perature decreases considerably with pressure from 300to 100 hPa.

Given a cloud distribution in the atmosphere, η iscalculated using (9) and the code for radiative transfer(Chou et al., 1995). Its results are displayed againstpressure in Figure 2 for three given cloud distributions.When the sky is clear, η > 1 above the 250 hPa level and

Figure 1. Vertical profiles of air temperature (thick line) and watervapour mixing ratio (thin line) obtained from averaging 52 days of

KWAJEX data.

η < 1 below that height. The factor η reaches a minimumof 0.89 at 428 hPa and a maximum of 1.8 at 100 hPa.Comparing Figures 1 and 2 shows that the height for themaximum η corresponds to the height for the minimumtemperature. Hence, η is largest at the tropopause.

Consider a thick stratiform cloud with the top at600 hPa. Since the cloud behaves like a blackbody, η

is close to unity in the cloud. Above the cloud, η varieswith height similar to in a clear sky. Nevertheless, η issmaller than that for a clear sky, because the upwardinfrared flux decreases with decreasing the underlyingblackbody temperature from the sea surface to cloud top.

Suppose the cloud top rises from 600 to 200 hPa.The cloud is so thick that it blocks the upward infraredradiation from the sea surface and the air below. Thus,it radiates upward as a blackbody but at a relatively lowcloud-top temperature. Hence, high cirrus significantlydecreases η and therefore greatly enhances the growth ofice crystals above itself.

2.4. The effect of crystal size and shape on Hic

Once εrη deviates from unity, the radiative effect on icecrystal growth is proportional to a ratio of (1 − Hic)/(1 −εrη). The ratio depends on crystal size and shape as wellas air temperature and pressure, which is discussed next.

Consider a spherical crystal. Given T and p, (1 −Hic)/(1 − εrη) varies with radius. Figure 3 displays theratio versus radius when fQ = 1. The ratio increases withcrystal size. Hence, Hic decreases and increases withcrystal size when η < 1 and η > 1, respectively, because



Figure 2. Infrared ratio η as a function of pressure. Thick, thin, anddashed lines represent cases with a clear sky, middle and high clouds,

respectively.

εr is close to unity (see section 5 for the effect of εr onHic).

The ratio (1 − Hic)/(1 − εrη) also varies with heightthrough air temperature and pressure. As shown inFigure 3, the ratio increases considerably with decreasingheight. For example, it increases from 59 to 81% at r∗ =1000 µm when (T , p) changes from (−80 °C, 100 hPa)to (−6 °C, 500 hPa).

Figure 3 is suitable for non-spherical crystals, too.A columnar-shaped crystal is modelled by a prolatespheroid of semi-major and minor axis lengths a andb, for which (Pruppacher and Klett, 1997)

C =√

a2 − b2/

ln{(

a +√

a2 − b2) /

b}

. (25)

Figure 3. Critical relative humidity Hic versus particle radius r∗, wherethick dot-dashed, thin dashed and solid lines represent the resultswhen (T , p) equals (−80 °C, 100 hPa), (−42 °C, 250 hPa) and (−6 °C,

500 hPa), respectively.

A plate-like ice crystal is modelled by an oblate spheroidof semi-major and minor axis lengths a and b, for which(Pruppacher and Klett, 1997)

C =√

a2 − b2/

arcsin√

1 − b2/a2. (26)

After defining r∗ in (4) and using it as the horizontalaxis, Figure 3 displays as Hic versus r∗ for non-sphericalcrystals.

3. Two unsteady modes

3.1. A linear model

Expression (24) is used to analyse how radiation impactsice crystal spectrum for a given relative humidity. Con-sider an ice crystal that deviates about its initial state;then (24) is linearized as

ds/dt = k(s0){Hi − Hic(s0)} + (s − s0)/τ(s0), (27)

where subscript ‘0’ indicates the initial state and

s = 4πr∗2 (28)

τ (s) = −{k(s)dHic/ds}−1 (29)

k(s) = 4πC

(1 − γ )2{HicA(Te) + B(Te)}(

dm

ds

)−1

. (30)

Buoyancy waves and underlying clouds modulate Hi

and Hic, respectively. Without loss of generality, supposethat

Hi − Hic(s0) = �Hi0 + AH sin(ωt), (31)

where the forcing parameters �Hi0, AH and ω areconstant. Substituting (31) into (27) and then solving theresulting equation yields

s − s0 = − kAH

τ−2 + ω2 [τ−1 sin(ωt) + ω{cos(ωt) − 1}]

+k�Hi0τ(et/τ − 1) + kAH(ωτ)

1 + (ωτ)2 τ(et/τ − 1)(32)

where k and τ take their values at s = s0. As shownin the preceding equation, crystal growth accelerateswhen AH > 0 and �Hi0 > 0, and shrinkage is enhancedwhen AH < 0 and �Hi0 < 0. These two unsteady modesbroaden and narrow the ice crystal spectrum whenη < 1 and η > 1, respectively, which is detailed insubsection 4.2.

The third term on the right-hand side of (32) representsthe bulk ice crystal growth in a period due to a buoyancywave or underlying-cloud variation. It has a factor of(ωτ)

/{1 + (ωτ)2}, which reaches a maximum at ωτ = 1.For a given τ , the factor (or the spectrum broadening) isdirectly proportional to the forcing period 2π/ω whenωτ � 1 and inversely proportional to the forcing periodwhen ωτ � 1.


614 X. ZENG

3.2. Time-scale for spectrum broadening

The second and third terms on the right-hand side of (32)share a common factor of τ(et/τ − 1) that is inverselyproportional to τ . For solid spherical crystals withoutinvolving radiation, τ → ∞, which results in τ(et/τ −1) = 0. Hence, an ice crystal spectrum with |τ | < ∞ isbroadened or narrowed with respect to a spectrum withτ → ∞. In brief, τ measures the spectrum broadening(or narrowing) due to radiation.

The time-scale τ relies on εr, η, crystal size, shape, airtemperature and pressure. As shown by (15), (18) and(29), τ is almost inversely proportional to 1 − εrη. Thus,the next discussions address τ(1 − εrη) versus crystalsize and shape first and then air temperature and pressure.

Figure 4 displays τ(1 − εrη) versus r∗ for five crystalshapes when the air temperature and pressure equal−80 °C and 100 hPa, respectively. Using ice crystalparameters from observations, the time-scale is calculatedwith (29), which is detailed next.

• A graupel particle is treated as a sphere with a bulkice density of 0.9 g kg−1 and ventilation factors fm

and fQ are assumed to be both unity. This crystalshape is so simple that τ versus r∗ is easily understoodwith (28)–(30). Thus, the shape is discussed here as areference for other shapes. Figure 4 displays the time-scale τ(1 − εrη) of graupel against r∗. The time-scaledecreases and increases with r∗ when r∗ is small andlarge, respectively. It reaches a minimum of 3.5 daysat r∗ = 5 µm.

• A bullet-shaped crystal is modelled with a prolatespheroid of semi-major and minor axis lengths a andb, for which the surface area is

S = 2πb

(b + a2√

a2 − b2arcsin

√a2 − b2

a

). (33)

The expressions for the mass and terminal velocityof bullets come from Heymsfield and Iaquinta (2000);the expression for the dimensional relationship fromHeymsfield (1972); and the expressions for the ventilationfactors from Hall and Pruppacher (1976). Using those

Figure 4. Time-scale τ(1 − εrη) versus particle radius r∗ for five crystalshapes when (T , p) equals (−80 °C, 100 hPa).

expressions, the time-scale for single bullets is calculatedand shown against r∗ in Figure 4. The curves of time-scale versus r∗ are not smooth because the curves forcrystal parameters (e.g. mass, dimensional relationship)versus size are not smooth either. Generally speaking,the time-scale curve for single bullets is similar to thatfor graupel except that τ(1 − εrη) reaches a minimum of3.3 days at r∗ = 12 µm.• A plate-like crystal is modelled with an oblate spheroid

of semi-major and minor axis lengths a and b, forwhich

S = 2πa

(a + b2√

a2 − b2ln

√a2 − b2 + a

b

). (34)

The mass of the plates with dendritic extensions iscalculated with the bulk density of Heymsfield (1972).The dimensional relationship of the plates is determinedwith the observations of Auer and Veal (1970). Theterminal velocity is calculated with the expression ofHeymsfield (1972). With those relationships, the time-scale for plate-like crystals is calculated and shownagainst r∗ in Figure 4. In contrast to graupel, plate-like crystals have a similar time-scale versus r∗ exceptthat τ(1 − εrη) reaches a minimum of 2.5 days at r∗ =8.9 µm when r∗ < 200 µm. The decrease in time-scale atr∗ = 200 µm coincides with the emergence of dendriticextensions at that size.• A column-like crystal is modelled with a prolate

spheroid. The expressions for the mass and termi-nal velocity of columns come from Heymsfield andIaquinta (2000), and that for the dimensional rela-tionship from Auer and Veal (1970) and Heymsfield(1972). Figure 4 displays the time-scale for column-like crystals against r∗ and shows that τ(1 − εrη) isclose to 3.6 days when r∗ is between 10 and 60 µm.

• A rosette crystal consists of bullets. It is modelledapproximately with a sphere. In the present paper,rosettes with four bullets are assumed unless specified.The mass and terminal velocity of rosettes are calcu-lated with the formulae of Heymsfield and Iaquinta(2000). Figure 4 also displays the time-scale for therosettes against r∗ and shows that τ(1 − εrη) reachesa minimum of 0.75 day at r∗ = 44.8 µm.

The time-scale for spectrum broadening, as summa-rized in Figure 4, varies with crystal shape. It varieswith crystal size similarly for different shapes. More-over, it varies considerably with height via air tempera-ture and pressure. Figure 5 displays the time-scale versusr∗ when (T , p) equals (−42 °C, 250 hPa) and (−6 °C,500 hPa), respectively. In comparison to Figure 4, thefigure shows that the time-scale versus r∗ is similar at dif-ferent heights, but the time-scale increases significantlywith height. These characteristics of the time-scale arehelpful in understanding the numerical results in the nextsection.



(a)

(b)

Figure 5. Same as in Figure 4 except that (T , p) equals (−42 °C,250 hPa) and (−6 °C, 500 hPa) in the upper and lower panels,

respectively.

4. Numerical simulations

The preceding linear model deals with the radiative effectwhile Hi is given. Since crystal growth consumes watervapour, Hi changes with time. Next, a nonlinear modelthat uses Hi as a prognostic variable is introduced to showthe radiative effect in a general framework.

4.1. A nonlinear model

Consider an air parcel that moves vertically. Based on theconservation of energy and water, its relative humidity isgoverned by

dHi

dt= Q1(Te)w − Q2(Te)

∑j

dmj

dt, (35)

where mj(j = 1, 2, . . .) denotes the mass of differentsized ice crystals, and

Q1(T ) = 1

T

(Lsg

RvCpT− g

Rd

)(36)

Q2(T ) = ρa

(RvT

Esi(T )+ RdL

2s

pRvCpT

). (37)

The expressions for Q1 and Q2 are derived similarlyto that for water vapour condensation in Rogers andYau (1989, p.106). Since the present crystal growth rate

takes account of the radiative effect, this parcel modelrepresents the radiative effect on ice crystal spectrum.

To model the spectrum evolution accurately, (24) ischanged to

da2

dt= 4πC(Hi − Hic)

(1 − γ )2{HicA(T ) + B(T )}(

dm

da2

)−1

, (38)

which uses a2 rather than a as a prognostic variablebecause the term on the right-hand side of (38) is almostconstant for spherical crystals. In the following numer-ical simulations, (35) and (38) are used as prognosticequations and are integrated with 1024 bins.

4.2. A mechanism for precipitation formation

Consider a stationary air parcel with εr = 1 and a constantη. Based on the linear model in the preceding section,the evolution of the ice crystal spectrum is reasoned asfollows. Since (1 − Hic)/(1 − η) increases with crystalsize, Hi changes so that eventually it lies between Hic

for small and large crystals, no matter what the initialrelative humidity is. When η < 1, Hi is lower than Hic

for small crystals but higher than that for large ones. Asa result, small crystals shrink due to sublimation withsome eventually disappearing. Large crystals grow dueto deposition with some of them becoming precipitatingcrystals. This sublimation–deposition process broadensthe ice crystal spectrum, and therefore converts cloud iceto precipitation. In contrast, when η > 1, Hi is higherthan Hic for small crystals but lower than that for largeones. As a result, large crystals shrink due to sublimationwhile small ones grow due to deposition, narrowing theice crystal spectrum.

To support this reasoning, a default numerical experi-ment is done to simulate plate-like crystal growth in anair parcel near the tropopause. Suppose that T = −80 °C,p = 100 hPa and the initial spectrum of ice crystals is

dN(a) = c1ae−c2ada, (39)

where N(a) denotes the number of ice crystals withthe semi-major axis length less than a. After choosingc1 = 1.25 × 1016 m−5 and c2 = 0.5 µm−1, the initialconcentration of ice crystals is 5 × 104 m−3 and theice water content 9 × 10−6 g m−3, close to observationsfor subvisual clouds (Heymsfield 1986). Given η = 0.5and the initial relative humidity Hi = 100%, the icecrystal spectrum and Hi are simulated using the timestep �t = 8 s. The modelled Hi is displayed against timein Figure 6. It decreases slowly with time. Meanwhile,the ice water content increases slowly due to waterconservation (figure omitted).

Figure 7 shows the evolution of the mass densitydM(a)/d ln(a) from t = 0 to 20 days, where M(a)

represents the mass of ice crystals with the semi-majoraxis length less than a. As shown in the figure, smallcrystals shrink due to sublimation and large ones growdue to deposition. Correspondingly, the median size of


616 X. ZENG

Figure 6. Relative humidity Hi versus time in the default experimentwhere T = −80 °C, p = 100 hPa and η = 0.5.

ice crystals increases with time. Since the small and largecrystals grow oppositely, Hi lies between the Hic forsmall and large crystals. Hence, Hi is close to the Hic forthe median size crystals. The increase in median crystalsize corresponds to the decrease in Hi shown in Figure 6.

If no radiation is involved (or η = 1), the ice crystalspectrum and Hi remain at their initial state. Hence,radiation is responsible for the spectrum broadeningshown in Figure 7.

To show the radiative effect when η > 1, the defaultexperiment is redone after increasing η from 0.5 to 1.5and using the final spectrum in Figure 7 as the initial one.Since (35) and (38) are symmetric with respect to η = 1,it is easily inferred that the spectrum evolution withη = 1.5 reverses that shown in Figure 7. Hence, largecrystals shrink due to sublimation and small ones growdue to deposition when η > 1. In other words, radiationnarrows the ice crystal spectrum when η > 1.

4.3. Sensitivity of precipitation formation to height

The preceding subsection reveals that precipitation canform on a time-scale of days at 100 hPa when η < 1.Here, two additional experiments with η = 0.5 are car-ried out to show that precipitation forms faster at lowerheights for the same η. The first additional experiment

Figure 7. Evolution of the mass density dM(a)/d ln(a) versus thehalf crystal size a in the default experiment where T = −80 °C andp = 100 hPa. The thick line denotes the initial spectrum, and the time

interval between lines is 1 day.

uses the same parameters as the default one exceptfor T = −42 °C and p = 250 hPa. In the experiment,a parameter for initial ice crystal spectrum c1 = 1.25 ×1018 m−5, which corresponds to an initial crystal con-centration of 5 × 106 m−3 and an ice water content of9 × 10−4 g m−3. The experiment is performed with atime step �t = 0.5 s. Figure 8 displays the evolution ofthe mass density from t = 0 to 6 hours, showing that pre-cipitation can form on a time-scale of hours at 250 hPa,much shorter than that at 100 hPa.

The second additional experiment uses the sameparameters as the first additional one except for T =−6 °C and p = 500 hPa. The experiment was performedwith �t = 0.02 s. Figure 9 displays the evolution of themass density from t = 0 to 15 minutes, revealing thatprecipitation can form on a time-scale of ten minutes atp = 500 hPa. In summary, precipitation formation dueto radiation is sensitive to height via air temperature andpressure. This sensitivity of precipitation formation toheight is a result of the sensitivity of the time-scale toheight shown in Figures 4 and 5.

5. Spectral asymmetry between absorbed andemitted radiation

Consider an optically thin cloud in a clear stationaryatmosphere. The cloud is so thin that it has almost

Figure 8. Same as in Figure 7 except that T = −42 °C, p = 250 hPaand the time interval between lines is 5 minutes.

Figure 9. Same as in Figure 7 except that T = −6 °C, p = 500 hPaand the time interval between lines is 1 minute.



no effect on atmospheric radiation. Thus, much of theinfrared radiation emitted by the underlying surface canreach the crystals in the cloud through the atmosphericthermal infrared window, leading to an obvious spectralasymmetry between crystal-absorbed and emitted radia-tion (or εr = 1). Next, a crystal in the cloud is discussedas an example to show the effect of the spectral asym-metry on Hic.

For an ice crystal in the cloud, εr is close to itsmaximum, or

εr ≈ ε0(r∗, Tust)/ε0(r

∗, T ) (40)

where Tust is the underlying surface temperature. Equa-tions (8) and (40) show that εr > 1 while T < Tust.

The infrared ratio η, just as shown in Figure 2, islarger than unity in the upper troposphere but smaller thanunity in the middle and lower troposphere. Consider anice crystal with an environment of η = 1.8, T = −80 °C,p = 100 hPa, and Tust = 300 K. Using εr in (40), Hic iscalculated and displayed against r∗ in Figure 10. Mean-while, the Hic with εr = 1 is displayed in the samefigure for comparison. Obviously, εr affects Hic con-siderably when r∗ < 40 µm. It changes Hic by 0.2% atr∗ = 12 µm. Since εr does not reverse the increase in Hic

with increasing r∗, it affects the sublimation–depositionprocess only quantitatively while η > 1.

In contrast, consider a crystal with an environment ofη = 0.9, T = −12 °C, p = 430 hPa and Tust = 300 K.Using εr in (40), Hic is calculated and displayed inFigure 11. It increases and decreases with r∗ while r∗is smaller and larger than r∗

MAX = 1.9 µm, respectively.In other words, Hic reaches a maximum of HicMAX =1 + 6 × 10−6% at r∗ = r∗

MAX. When 1 < Hi < HicMAX,there are two equilibrium states (or Hi = Hic(r

∗)) for anice crystal. The equilibrium state with the smaller r∗ isstable, corresponding to haze aloft. The other equilibriumstate with the larger r∗ is unstable. A crystal at theunstable equilibrium state, once perturbed initially, either

Figure 10. Critical relative humidity Hic as a function of particle radiusr∗, where η = 1.8, T = −80 °C, p = 100 hPa and Tust = 300 K. Thethin and thick lines represent Hic when εr equals one and is calculated

with (40), respectively.

Figure 11. Critical relative humidity Hic as a function of particle radiusr∗, where η = 0.9, T = −12 °C, p = 430 hPa and Tust = 300 K. Thethin and thick lines represent Hic when εr equals unity and is calculated

with (40), respectively.

shrinks to reach the stable state or grows to become aprecipitating crystal.

When many ice crystals coexist under η < 1 andsome of them have Hic less than 100%, the sublima-tion–deposition process works to form precipitation justas in the atmosphere with εr = 1. In summary, εr cannotchange the sublimation–deposition process greatly evenwhile η < 1.

6. Discussion

The preceding sections proposed a precipitation mech-anism: when η < 1, the sublimation–deposition processdue to radiation converts cloud ice to precipitating ice thatin turn sediments to dehumidify air. This process, differ-ent from particle sedimentation, can work as a source oflarge particles near cloud top. Next, cloud-top observa-tions are discussed to infer the existence of the process.

6.1. Subvisual clouds

Heymsfield (1986) observed the microphysical structureof subvisual clouds near the tropical tropopause andshowed that the clouds consist of ice particles with amean diameter of 2.5 µm and number density of lessthan 5 × 104 m−3. The subvisual clouds are common,but not pervasive, in the Tropics and subtropics (Winkerand Trepte, 1998). They are separated from cirrus belowbut tend to occur with thick cirrus below (Heymsfield,1986; Winker and Trepte, 1998). They can also persist forweeks or months in clear skies in the equatorial regions(Lynch and Sassen, 2002). These observations match thecharacteristics of the radiative effect, which is shownnext.

In the Tropics, convective clouds drive large-scaleupward motion in the troposphere (Riehl and Malkus,1958). Thus, horizontal water advection cannot generatesubvisual clouds near the tropopause. On the other hand,local vertical circulations (including turbulent motions)transport water downgradient near the tropopause. As aresult, water accumulates near the tropopause, and thuspervasive clouds are expected there.


618 X. ZENG

The sublimation–deposition process due to radiationcan break the pervasive clouds into subvisual clouds.Since high cirrus leads to η < 1 near the tropopause (seeFigure 2), the process, accompanied by high cirrus below,can convert small ice crystals to large ones that in turnsediment, serving as a water sink near the tropopause.Owing to the maximum η and minimum T at thetropopause, the conversion of cloud ice to precipitationis the least efficient there. As a result, the process canseparate subvisual clouds from cirrus below. With theaid of the sublimation–deposition process, high cirruscan enlarge the crystals near the tropopause that in turnradiate upwards strongly, which resembles the satellite-observed positive correlation between subvisual cloudsand the thick cirrus below.

In the absence of high cirrus, η > 1 near the tropopauseand large crystals shrink there, which can explain thesmall size of ice crystals in subvisual clouds. Since icecrystals shrink on a time-scale of 10 days (see Figures 4and 7), subvisual clouds can last for weeks or monthsin clear skies. In summary, the observations of subvisualclouds match the characteristics of the radiative effect,which suggests that the sublimation–deposition processexists near the tropopause.

6.2. Vertical distributions of ice crystal spectrum

Heymsfield et al. (2006) observed large crystals with ascale up to 103 µm at −85 °C. They reported that themaximum measured ice particle diameter increased from−70 to −85 °C when convection existed below. Thatobservational phenomenon, involving the origin of largecrystals right below the tropopause, can be explainedpartly by the radiative effect.

Suppose that convective cells eject ice crystals into theupper troposphere (Heymsfield et al., 2006), which arethen blown by the horizontal wind to form a stratiformcloud. The cloud in turn imposes a perturbation on theoriginal vertical η distribution. Consider, for example,a stratiform cloud in the atmosphere as discussed inFigure 1. When the cloud has a constant cloud ice mixingratio of 0.1 g/kg from 132 to 197 hPa, it introducesan η perturbation as shown in Figure 12. Since η < 1near cloud top, radiation can broaden the ice crystalspectrum there. In contrast, η > 1 near cloud base,which means that radiation narrows the spectrum there(Stephens, 1983). Thus, the radiative effect contributesto the increase in the maximum crystal size with height.

The stratiform cloud discussed in Figure 12 is opticallythick. It imposes a strong perturbation on the originalvertical η distribution so that η < 1 in a layer. However,an optically thin cloud introduces a weak η perturbationso that there is no layer with η < 1 and therefore no layerwith spectrum broadening. If convective clouds and theiranvils arise below a thin cloud, η above the convectiveclouds is decreased. As a result, η < 1 near the top of thethin cloud and η > 1 near the base, which contributes toan increase in the maximum ice crystal size with heightand partly explains the observational phenomenon that

Figure 12. Infrared ratio η as a function of pressure for two cases. Thethick line represents a case for a cloud (denoted by the shaded area).

The dashed line represents a case for a clear sky.

the maximum measured ice particle diameter increasedfrom −70 to −85 °C when convection existed below.

6.3. Cumulus cloud seeding

Precipitation initiation in warm cumulus clouds involvesthe formation of large drops on the scale of cloudlifetimes (e.g. 15 minutes). It requires some process toaccelerate the growth of drops at a radius of ∼20 µmwhere neither diffusional nor collectional growth issignificant. Current theories (e.g. giant salt particles, dropcollision due to turbulence, and cloud entrainment) stillinvolve some uncertainties (e.g. Rogers and Yau, 1989;Pruppacher and Klett, 1997), such that the precipitationinitiation is not yet fully understood. Based on the presentstudy, the radiative effect is proposed as a candidate toinitiate precipitation in warm cumulus clouds.

Consider a number of crystals in a blue sky. Theyform due to gravity waves (or other processes) and someof them have length greater than ∼4 µm. Since η < 1below the height of the 250 hPa level (see Figure 2),a part of the crystals can grow to become precipitatingones (see section 5) that fall and then melt below thefreezing level. Once the precipitating particles fall intocumulus clouds, they can serve as seeds to initiatefrequent collision between drops, which would in turngenerate precipitation quickly. From another perspective,radiation can broaden the cloud spectrum near cloudtop. Following the same procedure with (24) except forvariables for liquid water, an explicit expression for thecritical relative humidity with respect to water Hwc isobtained. Since Hwc functions in a similar manner as Hic

and η < 1 near cloud top, radiation can broaden the cloud



spectrum there and thus contribute partly to precipitationinitiation.

In the preceding three cases, the radiative effect isexpected to provide large particles near cloud top. Basedon the consistence between the characteristics of theradiative effect and the observations of subvisual clouds,it is inferred that the sublimation–deposition processexists in the upper troposphere. Using the present explicitexpression for ice crystal growth rate, the radiative effectwill be incorporated into a cloud-resolving model forfurther evaluation.

7. Conclusions

This theoretical study addresses the effect of radiationon ice crystal spectrum. From the heat balance equationof an ice crystal, an explicit expression for ice crystalgrowth rate is obtained that takes account of the radiativeeffect. The expression introduces three variables: thecritical relative humidity Hic, the infrared ratio η, andthe relative bulk absorption efficiency εr. An ice crystalgrows and shrinks when the relative humidity Hi > Hic

and Hi < Hic, respectively.The critical relative humidity Hic is associated with

η through an approximately linear relationship between(1 − Hic) and (1 − εrη) when the effect of solar radia-tion is ignored. The ratio (1 − Hic)/(1 − εrη) depends oncrystal size and shape as well as air temperature and pres-sure. Generally speaking, the ratio increases with crystalsize. Thus, given εr = 1, Hic increases and decreases withcrystal size when η > 1 and η < 1, respectively.

A linear model is introduced to show how radiationbroadens (or narrows) an ice crystal spectrum, wherethe time-scale for spectrum broadening relies on εr, η,crystal size, shape, air temperature and pressure. Thetime-scale is computed using observed parameters of icecrystals. It increases considerably with height through airtemperature and pressure. Given η = 0.5, the time-scaleis measured in minutes, hours and days at the 500, 250and 100 hPa levels, respectively.

A nonlinear model that uses Hi as a prognostic variableis introduced to simulate the evolution of ice crystalspectrum due to radiation. Since Hic varies with crystalsize, water is redistributed between ice crystals withdifferent sizes. When η < 1, smaller ice crystals shrinkdue to sublimation and larger ones grow due to vapourdeposition, which results in precipitation formation evenwhen ice water content is small and air is stationary.

The infrared ratio η approximately measures the rela-tive magnitude of inward and outward infrared fluxes onthe surface of an ice crystal. Owing to the great influ-ence of clouds on η, the precipitation formation dueto radiation is associated with clouds below. Based onthe consistence between the characteristics of the radia-tive effect and the observations of subvisual clouds, it isinferred that the precipitation formation due to radiationexists in the upper troposphere.

However, the present parcel model needs to incorporatemore physical processes so that its results are quantita-tively comparable to field observations. Introducing icecrystal sedimentation, for example, is important in mod-elling ice crystal spectrum (e.g. Jensen et al., 2007). Aspectrally-resolved treatment of radiation (e.g. Stephens,1983; Wu et al., 2000) is important in the accurate deter-mination of the infrared ratio η, the relative bulk absorp-tion efficiency εr, and the absorbed solar flux Fs. Withthe full representation of the physics, the model can sim-ulate optically-thin clouds so that its results are close tofield observations.

Acknowledgements

This study is supported by the Office of Science, USDepartment of Energy through the Atmospheric Radia-tion Measurement (ARM) Program. The author appre-ciates Andy Heymsfield, Graeme Stephens, David O’C.Starr, Norihiko Fukuta and Ken Beard for clarifyingmany issues on cloud microphysics and radiation in thecourse of this study. He thanks Wei-Kuo Tao for kindsupport and Steve Lang for improving the manuscript.Special thanks are extended to two anonymous reviewersfor their critical yet constructive comments.

Appendix A List of Symbols

a/b : semi-major/semi-minor axis lengthC : stationary diffusion shape factorCp : specific heat of dry airD : diffusivity of water vapoure : partial pressure of water vapourec : partial pressure of water vapour in the

critical case, see (12)Esi : saturation vapour pressure over icefm/fQ : ventilation factor for mass transfer/

thermal diffusionFs : solar flux absorbed by an ice crystalF +/F − : upward/downward infrared fluxFα/Fβ : factor for the kinetic effect in heat/water

vapour diffusiong : acceleration due to gravityHi = e/Esi : relative humidity with respect to iceHic : critical relative humidity, see (18)K : coefficient of thermal conductivity of

dry airLs : latent heat of sublimationm : mass of an ice crystalM(a) : mass of ice crystals with semi-major

axis length shorter than a

N(a) : number of ice crystals with semi-majoraxis length shorter than a

r∗ = S/4πC : equivalent radius of an ice crystal, see(4)

Rd/Rv : gas constant for dry air/water vapourp : air pressurepo = 105 Pa : reference pressure


620 X. ZENG

s = 4πr∗2 : area variable for modelling, see (28)S : surface area of an ice crystalt : timeT : (air) temperatureTe : environmental temperature around an

ice crystalTo : absolute temperature of ice pointTs : surface temperature of an ice crystalTsc : crystal surface temperature in the crit-

ical case, see (11)α/β : thermal accommodation/deposition

coefficientγ : relative change of crystal surface tem-

perature due to radiation in the criticalcase, see (13)

ε0 : bulk absorption efficiency of an icecrystal for blackbody radiation

εr : relative bulk absorption efficiency ofan ice crystal for infrared radiation

η : infrared ratio, see (9)ρa/ρv : density of air/water vapourρsi : saturation density of water vapour

over iceσ : the Stefan–Boltzmann constantτ : time-scale for spectral broadening due

to radiation, see (29)χ : ratio of the infrared flux to the conduc-

tive heat flux from air to an ice crystalwhile both air and the crystal are treatedas blackbodies

�t : time step for numerical integration

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the inï¬‚uence of radiation on ice crystal spectrum in the

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