NASA/TM-1999-104606, Vol. 15

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Volume 15

A Solar Radiation Parameterization for

Atmospheric Studies

Ming-Dah Chou and Max J. Suarez

Climate and Radiation Branch

NASA Seasonal to Interannual Prediction Project

June 1999
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NASA/TM-1999-104606, Vol. 15

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Goddard Space Flight Center, Greenbelt, Maryland

Volume 15

A Solar Radiation Parameterization for

Atmospheric Studies

Ming-Dah Chou

Max J. Suarez

Goddard Space Flight Center

Greenbelt, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

June 1999
NASA Center for AeroSpace Information

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Hanover, MD 21076-1320

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Price Code: A10
Abstract

This NASA TechnicalMemorandumdocumentsthe solarradiativetransfermodel

(CLIRAD-SW)developedat the GoddardClimateand RadiationBranch,which hasbeen

appliedto variousatmosphericmodelsusedin the GoddaradLaboratoryfor Atmospheres,

includinga generalcirculationmodel,a mesoscalemodel,anda cloudensemblemodel. It

includestheabsorptiondueto watervapor,03,02,CO2,clouds,and aerosols.Interactions

amongtheabsorptionandscatteringby clouds,aerosols,molecules(Rayleighscattering),and

the surfacearefully takeninto account. Fluxesare integratedvirtually over the entire

spectrum,from 0.175 pm to 10 pm.

Depending upon the nature of absorption, different approaches are applied to different

absorbers. In the ultraviolet (UV) and photosynthetically active (PAR) region, the spectrum is

divided into 8 bands, and single 03 absorption coefficient and Rayleigh scattering coefficient

are used for each band. In the infrared, the spectrum is divided into 3 bands, and the kdistribution method is applied. Ten absorption coefficients are used in each band. The flux

reduction due to 02 is derived from a simple function, while the flux reduction due to CO2 is

derived from precomputed tables.

Reflection and transmission of a cloud and aerosol-laden layer are computed using the

5-Eddington approximation. Fluxes are then computed using the two-stream adding

approximation. For a cloud layer, the optical thickness is parameterized as a function of

cloud water/ice amount and the effective particle size, whereas the single-scattering albedo and

asymmetry factor are parameterized as a function of the effective particle size. The

parameterizations are applied separately to water and ice particles. A maximum-random

approximation is adopted for the overlapping of clouds at different heights. Aerosol optical

properties are specified input parameters to the radiation model.

A special feature of this model is that absorption due to a number of minor absorption

bands is included. Individually the absorption in those minor bands is small, but collectively

the effect is large, ~10 % of the atmospheric heating. Integrated over all spectral bands and

all absorbers, the surface heating is computed accurately to within a few watts per meter

squared of high spectral-resolution calculations, and the atmospheric heating rate between

0.01 hPa and the surface is accurate to within 5%.

A complete computer code of the CLIRAD-SW model with sample calculations is

available at http://climate.gsfc.nasa.gov/-chou/clirad_sw

iii
iv
TABLE OF CONTENTS

LIST OF FIGURES ..........................................

LIST OF TABLES ...........................................

1. INTRODUCTION .........................................

2. SPECTRAL BANDS ........................................

3. GASEOUS ABSORPTION ...................................

3.1 High spectral-resolution calculations ...........................

3.2 The k-distribution method ..................................

3.3 Absorption due to water vapor in the infrared (Bands 9-11) ...........

3.4 Absorption due to water vapor in the PAR region (Band 8) ............

3.5 Absorption due to 0 3 in the UV and PAR regions (Bands 1-8) ..........

3.6 Absorption due to 0 3 in the infrared ...........................

3.7 Absorptwn due to 0 2 ......................................

3.8 Absorption due to CO 2 .....................................

4. MOLECULAR AND PARTICULATE SCATTERING ................

4.1. Rayleigh scattering ......................................

4.2. Cloud and rain single-scattering properties .....................

4.3. Aerosol single-scattering properties ..........................

5. SURFACE ALBEDO .......................................

6. FLUX CALCULATIONS ...................................

6.1 Layer optical properties ....................................

6.2 Layer reflectivities and transmissivities ........................

6.3 Two-stream adding approximation and flux calculations .............

6.4 Flux reduction due to CO 2 and 0 2 ..............................

7. CLOUD OVERLAPPING ....................................

7.1. Maximum-random overlapping ..............................

vii

ix

1

1

3

3

3

6

8

9

10

10

12

14

14

14

17

21

21

21

22

23

25

25

26

V
7.2. Scaling of opticalthickness ................................

8. SAMPLE CALCULATIONS ..................................

8.1. Effect ofabsorption and scattering in minor bands ................

8.2. Clear-skyfluxes and heating rates ............................

8.3. Cloudy-skyfluxes and heating rates ...........................

ACKNOWLEDGMENTS ......................................

REFERENCES .............................................

28

31

31

31

34

35

36

vi
List of Figures

. Spectral distribution of the transmission function of the entire column of a midlatitude summer atmosphere in the vertical direction averaged over 10 cm 1 for water

vapor, 03, 02, and CO2 ......................................... 7

. The effect of CO2 absorption in the spectral regions 2200-2400 cm 1 and 3500-3760

cm _ on the solar heating. Overlapping with water vapor absorption is included in the

calculations of CO2 effect. The solid curve is line-by-line calculations, and the dashed

curve is the parameterization using the precomputed table of &F(p C/I.lo,P) given by

Equation (3.19). The CO2 concentration is set to 350 ppmv, and the solar zenith

angle is set to 60 ° . ........................................... 12

. The extinction coefficient of ice and water clouds as a function of the effective

particle size. The values of ri and r, are, respectively, the effective size of ice and

water particles. The data points are derived from Equation (4.4), whereas the curves

are the regression using Equation (4.6) ............................ 18

. The single-scattering co-albedo of ice and water clouds as a function of the effective

particle size. The values of ri and r, are, respectively, the effective size of ice and

water particles. The data points are derived from Equations (4.1)-(4.3), whereas the

curves are the regression using Equation (4.7) ....................... 20

. The asymmetry factor of ice and water clouds as a function of the effective particle

size. The values of ri and r, are, respectively, the effective size of ice and water

particles. The data points are derived from Equation (4.5), whereas the curves are the

regression using Equation (4.8) ................................... 20

. The left panel is the diffuse reflectivity computed from (6.5) minus that computed

from (6.7). The right panel is the diffuse transmissivity computed from (6.6) minus

that computed from (6.8). The cloud asymmetry factor is set to 0.85 ....... 23

vii
.

.

.

10.

The maximum-random cloud-overlapping scheme. Clouds are identified as high,

middle, and low separated roughly by the 400 hPa and 700 hPa levels. Clouds are

assumed maximally overlapped within each height group and randomly overlapped

among different height groups (left panel). Scaling of the optical thickness applies

only to the maximally-overlapped cloud layers (right panel) ........... 27

m

Scaling functions for the cloud optical-thickness, Z and Z, for direct (shown only for

Po = 0.3) and diffuse radiation, respectively. The value of f' is the normalized cloud

cover defined by (7.8). Numbers in the figure are the cloud optical thickness. The

dots are Z = f' 1.5 used in the NCAR CCM model (Briegleb, 1992) ....... 30

The heating rate profiles of a mid-latitude summer atmosphere computed with the

line-by-line method (solid curves) and the parameterization (dashed curves). The

heating is express as a function of pressure in the upper panel but as a function of the

logarithm of pressure in the lower panel. The solar zenith angle is set to 60 °, surface

albedo is set to 0.2, and the CO2 concentration is set to 350 ppmv. Absorption due to

water vapor, 03, 02, and CO2 are included. Scattering is excluded ........ 3 3

Same as Figure 9, except a cloud deck is inserted between 800 and 920 hPa. The

cloud optical thickness is 9.7 in the visible region, and the effective radius of cloud

liquid drops is 12 pm ........................................ 35

Vlll
List of Tables

. Gaseous absorption and Rayleigh scattering in the broadband radiation

parameterizations and the equations used for transmittance and flux calculations.

Clouds and aerosols are included in all bands ..................... 2

. The flux-weighted k-distribution function of water vapor, h, in the infrared for

Pr=300 hPa and 0,.=240 K. Sum of all h values is 0.52926. The unit of the

absorption coefficient, k, is g l cm 2 .............................. 8

. The spectral range, fractional solar flux, effective ozone absorption coefficient, and

Rayleigh scattering coefficient. Sum of S/S O is 1 .................. 9

. Coefficients ao and aI of the parameterization, Equation (4.6), for the cloud

extinction coefficient The units of a o and aI are, respectively, m 2 g 1 and m 2 g _ pm

.......................................................... 8

. Coefficients b0, ba, and b 2 of the parameterization, Equation (4.7), for the cloud

single-scattering co-albedo. The units of b I and b2 are, respectively, _am_ and _am2 . .

.......................................................... 9

. Coefficients Co, cland C2 of the parameterization, Equation (4.8), for the cloud

asymmetry factor. The units of ca and c2 are, respectively, _am _ and _am 2 ... 19

. Effects of minor absorption and scattering on the downward surface solar flux from

detailed high spectral-resolution calculations and parameterizations. Fluxes are

computed for a typical mid-latitude summer atmosphere. The CO2 concentration is

set to 350 ppmv. The solar zenith angle and the surface albedo are set to 60 ° and

zero, respectively. Units are W m 2 ............................. 32

. Net downward solar fluxes at the top of the atmosphere (TOA) and at the surface

(SFC) and the radiation absorbed in the atmosphere (ATM) computed using high

spectral resolution calculations and the parameterization. The atmosphere used is

typical of a mid-latitude summer with a CO2 concentration of 350 ppmv. The solar

zenith angle and the surface albedo are set to 60 ° and 0.2, respectively. Calculations

include the absorption due to water vapor, 03, CO2 and 02 covering the spectral range

from 0.175 _am to 10 _am. Scattering is excluded. Units are W m 2 ..... 33

ix
. Same as Table 8 except scattering by gases and cloud is included. A deck of stratus

cloud with an optical thickness of 9.7 in the visible region is located between 800 and

920 hPa. Units are W m 2 35
1. INTRODUCTION

Solar radiation is the ultimate source of energy for the Earth's climate. A small error

in heating rate calculations might induce large errors in model-simulated climate. It is very

important that solar heating is calculated accurately in an atmospheric model, not only for the

total heating but also for the horizontal and vertical distributions of the heating. One of the

major difficulties in the calculation of atmospheric radiative heating is that it requires a huge

amount of computing time using detailed spectral and angular integrations. There are tens of

thousands of molecular lines across the spectrum. The half-width of a line is ~0.05 cm 1 in

the troposphere, which is much narrower than the mean spacing of lines. As a result, the

absorption coefficient varies rapidly with wavenumbers. Detailed spectral integration of

radiative transfer requires >106 spectral points. When scattering due to clouds and aerosols is

involved, angular integration becomes the major computational burden. Adding to the

complexity of the radiative transfer calculations are the overlapping of absorption due to

multiple absorbers, the overlapping of clouds at various heights, and the wide ranges of cloud

microphysical and optical properties.

There are several solar radiation parameterizations with various complexities available

for use in atmospheric models (Fouquart et al., 1991 and the references therein). At the

NASA/Goddard Space Flight Center, Climate and Radiation Branch, we have developed a

model for solar radiative transfer, which can be traced back at least to the past 10 years.

Depending upon the spectral properties of absorbers and scatterers, as well as the heights

where heating is important, various parameterizations are applied. This model has been

applied to the various atmospheric model used in the Goddarad Laboratory for Atmospheres,

which include a general circulation model (Koster and Suarez, 1995), a mesoscale model, and

a cloud ensemble model (Tao et al., 1996). This NASA Technical Memorandum documents

the detailed treatment of the solar radiative transfer in the model.

2. SPECTRAL BANDS

Table 1 shows the spectral bands of the solar radiation model. There are eight bands in

the ultraviolet and visible region (v>14280 cm 1) and three bands in the infrared region

(v<14280 cm 1). Also shown in the table are the absorbers and scatterers included in the

calculation of solar fluxes in each band, as well as the equations used for computing the
Table 1. Gaseousabsorptionand Rayleighscatteringin the broadbandradiation

parameterizations andthe equationsusedfor transmittanceandflux calculations.

Cloudsandaerosolsareincludedinallbands.

Band SpectralRange Absorber/ Equation

Scatterer

(era 1) (/am)

1

2

3

4

5

6

7

8

9

10

11

(44440-57140)

(40820-44440

35700-38460)

(38460-40820)

(33900-35700)

(32260-33900)

(31250-32260)

(25000-31250)

(14280-25000)

(8200-14280)

(4400-8200)

(1000-4400)

Total Spectrum

Total Spectrum

(0.175-0.225) 03 3.12

Rayleigh --

(0.225-0.245 03 3.12

0.260-0.280) Rayleigh --

(0.245-0.260) 03 3.12

Rayleigh --

(0.280-0.295) 03 3.12

Rayleigh --

(0.295-0.310) 03 3.12

Rayleigh --

(0.310-0.320) 03 3.12

Rayleigh --

(0.320-0.400) 03 3.12

Rayleigh --

(0.400-0.700) 03 3.12

H20 3.12

Rayleigh --

(0.70-1.22) H20 3.11

03* 3.12

Rayleigh --

(1.22-2.27) H20 3.11

Rayleigh --

(2.27-10.0) H20 3.11

02 3.16

CO2 3.19; 3.24

* 03 absorption is folded into Band 8
transmissionfunction. The first eight bandsinvolves the 03 absorptionand Rayleigh

scattering.Band8 alsoincludestheweakabsorptiondueto watervapor. Singlevaluesof the

03 absorptioncoefficient,the watervapor absorptioncoefficient,and Rayleighscattering

coefficientareusedin eachof the8 bands.Bands9-11 includethe watervaporabsorption

andRayleighscattering.Watervaporabsorptionin thesebandsis significant,and the kdistribution method is used. Band 9 also includes the weak 03 absorption. The 03 absorption

in this band is folded into the absorption in Band 8. The absorption due to 02 and CO2 is of

secondary importance but occur in wide spectral ranges. Different approaches which

compute only the reduction in fluxes are used. Clouds and aerosols are included in all bands.

3. GASEOUS ABSORPTION

3.1 High spectral-resolution calculations

The absorption coefficients of water vapor, 02, and CO2 vary rapidly with wavenumber.

Parameterizations for the absorption due to these gases are based on line-by-line calculations

using the latest (1996) version of the molecular line parameters compiled at AFGL (Rothman,

1987). The shape of an absorption line is assumed to follow the Voigt function. The Voigt

half-width is a function of pressure, temperature and wavenumber. Because the shape of the

far wings of a line is not well known, the absorption coefficient of a molecular line is set to

zero at wavenumbers >10 cm 1 from the line center. The spectral resolution of the line-byline calculations is 0.002 cm 1, which is adequate for solar flux calculations. The 03

absorption coefficient varies by several orders of magnitude in the UV and visible regions but

rather smoothly with wavenumber. Absorption due to 03 is computed with a resolution of

~0.003 pro. The absorption coefficient is interpolated from the spectral values given in WMO

(1985). The effect of pressure and temperature on the 03 absorption is small and, hence,

neglected.

3.2 The k-distribution method

The width of an absorption line of water vapor, 02 , or CO2 is much smaller than the

average spacing between absorption lines. It decreases linearly with decreasing pressure and

is only ~0.05 cm 1 in the lower troposphere. As a result, the absorption coefficient varies

rapidly within narrow spectral intervals, and accurate radiative transfer calculations require a
highspectralresolution.Forthecaseof watervaporabsorption,thehigh resolutionline-bylinemethodrequiresflux calculationsat >10 6 spectral points. For an atmospheric layer where

temperature 0 and pressure p can be assumed constant, the wavenumbers with the same

absorption coefficient are radiatively identical and can be treated as one entity. Fluxes at

those wavenumbers with the same absorption coefficient need to be computed only once.

Therefore, flux calculations can be greatly simplified by grouping the wavenumbers with the

same absorption coefficient. Within a small interval _V where the spectral variation of

insolation (incoming solar radiation) is small, the integration of fluxes over wavenumbers can

be replaced by that over the absorption coefficient. The mean transmission function of the

interval _V can be written as

oo //

T(w') = f e-G(P'°_'dv / _v= f e-k_'g(k;p,O)dk = Zai(p,O)e -ki_' (3.1)

_v 0 i=1

where w' is the absorber amount, k v is the absorption coefficient at the wavenumber v, g is

the k-distribution density function (cf. Arking and Grossman, 1972) such that the fraction of

spectrum with the absorption coefficient between k - dk ] 2 and k + dk ] 2 is g(k)dk, and ai

[=g(ki)Aki] is the k-distribution function. Chou and Lee (1996) has shown that the kdistribution method requires only 10 values of k (i.e. n=10) for accurate calculations of

solar heating due to water vapor using the k-distribution method, instead of >10 6 spectral

points using the line-by-line method.

In the atmosphere where pressure and temperature change with height, the

wavenumbers with a common absorption coefficient at a given height will not necessarily

have a common absorption coefficient at other heights. These wavenumbers are no longer

radiatively identical, and the k-distribution method cannot be used without applying

assumptions. Pressure affects the width of a molecular line. Near the center of a line, the

absorption is strong, and flux calculations are not sensitive to errors in the absorption

coefficient. Distant from line centers, the absorption coefficient varies nearly linearly with

pressure. On the other hand, temperature affects absorption primarily through its effect on

the line intensity, which is constant for a given line. Thus, the temperature effect on

absorption is rather smooth with respect to wavenumber. It has been shown by Chou (1986)

that fluxes can be calculated accurately using the one-parameter scaling for the absorption

coefficient,

4
kv(p,O ) = kv(Pr,Or) f(O, Or) (3.2)

where Pr is the reference pressure, 0 r is the reference temperature, m < 1, and f(O, Or) is

the temperature scaling function. As explained in Chou (1986), the optical path between the

top of the atmosphere and the middle and lower troposphere is large near the center of

absorption lines. It follows that solar fluxes in the middle and lower troposphere are

primarily attributable to the spectral intervals between absorption lines, and the wingapproximation of (3.2) can be applied to accurately compute fluxes. In the stratosphere, the

solar heating due to water vapor and 02 is small, and underestimation of the absorption

coefficient induced by the use of (3.2) does not have a significant effect on the heating rate.

In the upper troposphere, the absorption is neither strong nor weak, and the error induced by

the scaling is expected to have a large impact on flux calculations. However, if we choose Pr

to be the upper tropospheric pressure, then p/Pr = 1 in the upper troposphere, and the

scaling introduces only a small error in the absorption coefficient. It has been found (Chou,

1986) that the solar fluxes can be computed accurately by choosing Pr=300 hPa and

m=0.8. Flux calculations are not sensitivity to m for 0.7<m<0.9. The effect of temperature

on the absorption of solar radiation is weak.

approximated by

f(O, 0r) = 1 + 0.00135(8 - 0r)

For the absorption due to water vapor, it is

(3.3)

where 0r=240 K. For the absorption due to 02, and CO2, the temperature effect is neglected,

i.e. f(O, Or)=l.

Using the scaling of (3.2), Equation (3.1) becomes

//

T(w) = _],ai(Pr, Or)e -k;"

i=1

(3.4)

where

w(p,O) = w' f(O, Or) (3.5)

Thus, the scaling of the absorption coefficient is reduced to the scaling of the absorber

amount. The mean transmission given in (3.4) is independent of wavenumber, and flux

calculations are greatly simplified. It is noted that the physical basis for the k-distribution
method with the one-parameter pressure and temperature scaling is the same as for the

correlated k-distribution method (Wang and Ryan, 1983; Goody et al., 1989; Lacis and

Oinas, 1991; Fu and Liou, 1992; Mlawer et al., 1997), i.e., the dominant effect of line wings

on the absorption. The correlated k-distribution method correlates the values of k at

different pressure and temperature through table look-up. The k-distribution method with

the one-parameter scaling correlates values of k through (3.2). The accuracies of these two

approaches are comparable, but the latter is much simpler than the former.

3.3 Absorption due to water vapor in the infrared (Bands 9-11)

Figure la shows the spectral distribution of the water vapor transmission function of the

entire atmospheric column in the vertical direction averaged over 10 cm 1. The atmosphere

used is typical of mid-latitude summer conditions (Anderson et al., 1986). It shows that water

vapor absorption varies strongly with wavenumber. The k-distribution method is used to

parameterize water vapor absorption. For a spectral band with a width AV, the direct solar

flux at a given pressure level p can be written as

F(w) = Ie-kv_S, dv

Av

(3.6)

where S v is the extraterrestrial solar flux density at the wavenumber v multiplied by the

cosine of the solar zenith angle, k is the absorption coefficient at Pr and /9,., and w is the

scaled absorber amount, given by (3.5), above the pressure level p and in the direction of the

solar beam. The flux-weighted mean transmission of a band is given by

T(w)= Ie-L''S, dv/ IS, dr (3.7)

kv kv

If we divide a spectral band into m small intervals and apply the k-distribution method

of (3.4), Equation (3.6) becomes

F(w) = Z Sj e-k"_ dv = e -ki_

jl= LA,i j i=ILj=I l

(3.8)

where Sj is the mean flux density at the top of the atmosphere in the interval AVj, ai, j is the

k-distribution function in the spectral interval AVj, n is number of k-intervals, and
1.0

0.8

0.6

E 0.4

0.2

I"--

0.0

0

1.00

Q_

o 0.95

"_ 0.90

0.85

I--

0.80

10000

a) H O

5000 10000 15000 20000

b) O

II

15000 20000 25000 30000

Wavenumber (cm _)

1.0

0.9

0.8

0.7

0.6

0.5

6000 6000

d) CO

1.0

rl.....

0.6 t

0.6

0.4

0.2

0.0 ,,

2000 3000

c) O

, , ,,, , , , , , , , , ,

V

A

, , , i , , , i , , , i ,

10000 12000

BI --

, i , , , -

14000 16000

, i , , i .... i

4000 5000 6000 7000 8000

Wavenumber (cm _)

Figure 1. Spectral distribution of the transmission function of the entire column of a mid-latitude summer

atmosphere in the vertical direction averaged over 10 cm 1for water vapor, 03, 02, and CO>

m

Av = Z Avj (3.9)

j=l

By defining

m

h = Zai,jSjAvj / SO

j=l

(3.10)

which is the flux-weighted k-distribution function at pr=300 hPa and 0r=240 K, Equation

(3.8) becomes

F(w) -k,.

= _o 2..a hi8

/=1

(3.11)

where S o is the total insolation at the top of the atmosphere (=extraterrestrial solar flux

multiplied by the cosine of the solar zenith angle). Equation (3.11) is applied to compute the
watervaportransmissionfunctionin the infrared(Bands9-11). Fluxesarecomputedfor

eachki, and the total flux is the sum of these fluxes weighted by the k-distribution function.

Table 2 gives the values of k and the k-distribution function used in the three IR bands. It is

noted that if n is smaller than 10, say, 7, the calculated heating rate profile will oscillate

strongly with height, as shown in Figure 1 of Chou and Lee (1996). It is also noted that the

reason for dividing the infrared into three bands is the large change with wavenumber in the

cloud and aerosol optical thickness and single-scattering albedo.

Table 2. The flux-weighted k-distribution function of water vapor, h, in the infrared for Pr =300 hPa and

Or =240 K. Sum of all h values is 0.52926. The unit of the absorption coefficient, k, is g 1cm 2.

k-Interval k h (Band 9) h (Band 10) h (Band 11)

1 0.0010 0.20673 0.08236 0.01074

2 0.0133 0.03497 0.01157 0.00360

3 0.0422 0.03011 0.01133 0.00411

4 0.1334 0.02260 0.01143 0.00421

5 0.4217 0.01336 0.01240 0.00389

6 1.3340 0.00696 0.01258 0.00326

7 5.6230 0.00441 0.01381 0.00499

8 31.620 0.00115 0.00650 0.00465

9 177.80 0.00026 0.00244 0.00245

10 1000.0 0.00000 0.00094 0.00145

3.4 Absorption due to water vapor in the PAR region (Band 8)

The absorption due to water vapor in the photosynthetically active radiation (PAR)

region between 14280 cm 1 and 25000 cm _ (Band 8) is weak (Figure la), and the mean

transmission defined by (3.7) can be approximated by

T(w)_ I(1-kvw)Svdv/ ISvdv:l--kw=e-A (3.12)

kv kv

where k is an effective absorption coefficient, and w is the scaled water vapor amount given

by (3.5). Using the line-by-line method, the mean transmission function T(w) given by

(3.7) is first computed at 300 hPa and 240 K for a large range of w. For each w, an

effective absorption coefficient is then computed by inverting (3.12),

k(w)=-l--lnT(w)

W

(3.13)
Thevalueof k is found to range from 0.00065 to 0.00080 g l cm 2, and a mean value of

0.00075 g 1 cm 2 is used to compute the water vapor absorption in Band 8 from (3.12).

3.5 Absorption due to 0 3 in the UV and PAR regions (Bands 1-8)

The eight bands in UV and PAR spectral regions as shown in Table 1 are relatively

narrow. The range of the 03 absorption coefficient in each band is small and the mean

transmission function can also be approximated by (3.12). By dividing the UV and PAR

regions into 127 narrow bands with a width of ~0.003 pm and using the 03 absorption

coefficient given in WMO (1986), the mean 03 transmission function is computed from (3.7).

The same procedures described above for the determination of the effective water vapor

absorption coefficient is applied to derive the effective 03 absorption coefficient, which is

given in Table 3. It is noted that the effect of temperature and pressure on the 03 absorption

Table 3. The spectral range, fractional solar flux, effective ozone and water vapor absorption coefficients,

and Rayleigh scattering coefficient. Sum of S / S O is 1.

Band S/S O 03 Absorption H20 Absorption Ray. Scattering

Coefficient Coefficient Coefficient

(cm_atm) 1 (gl cm 2) (hPa 1)

UV-C

UV-B

UV-A

PAR

Near IR

1 0.00057 30.47 0.00604

2 0.00367 187.24 0.00170

3 0.00083 301.92 0.00222

4 0.00417 42.83 0.00132

5 0.00600 7.09 0.00107

6 0.00556 1.25 0.00091

7 0.05913 0.0345 0.00055

8 0.39081 0.0572 0.00075 0.00012

9 0.32055 See Table 2 0.0000156

10 0.16536 See Table 2 0.0000018

11 0.04335 See Table 2 0.0

* Enhanced by the 03 absorption in Band 9.
is small,andthe one-parameterscalingis not appliedto the03amount.Also shownin the

tableis the ratio of the extraterrestrialsolarradiationof eachband to that of the total

spectrum,S/S o. Equation (3.12) is applied to compute the transmission due to 03 in Bands

1-8.

3.6 Absorption due to 0 3 in the infrared

As shown in Figure lb, the absorption due to ozone in the near infrared (Band 9) is

restricted to a narrow spectral region next to Band 8 where absorption due to water vapor is

weak. However, the absorption due to water vapor in the rest of Band 9 is not necessarily

weak. Thus, calculations of the ozone heating in the near infrared would require adding

another band in the broadband radiation model. Because the absorption due to water vapor

and 03 is weak in Band 8, the 03 absorption in Band 9 can be folded into the absorption in

Band 8 as if it were the absorption due to another absorber. To do so, the effective 03

absorption coefficient of Band 8, k 8 , is replaced by

k = k s +Ak (3.14)

where Ak satisfies

(1-e -Ak_') ISvdv= I(1-e-kv_')Svdv

Av 8 Av 9

(3.15)

so that the absorption of solar radiation due to 0 3 in the near infrared is correctly computed,

where w' is the ozone amount, and Av 8 and Av 9 are the widths of Bands 8 and 9,

respectively. For a wide range of w' found in the atmosphere, the value of Ak ranges

between 0.0032 and 0.0033 (cm-atm)sTp 1. Therefore, the ozone absorption coefficient in

Band 8 is enhanced by 0.0033 (cm-atm)sTp 1 to take into account the absorption by ozone in

the infrared. The 03 absorption coefficient of Band 8 given in Table 3 includes this extra

absorption. This approach to computing the ozone absorption in the near infrared requires

no extra computing time.

3.7 Absorption due to 0 2

The absorption due to 02 occurs in narrow spectral intervals, but is not necessary weak

near band centers (Figure lc). Using the line-by-line method, Chou (1990) computed the

10
mean 02 transmission function in the A and B bands centered at 13150 cm 1 and 14510 cm 1

and fit the transmission function by a square-root approximation,

T(w) = e -;4_ (3.16)

The absorption in the weak bands centered at 7890 cm 1 and 15870 cm 1 was not

included. To include the absorption in all those bands, the mean transmission function of 02

at Pr=300 hPa and 0,.=240 K in the spectral regions 7600-8050, 12850-13190, 14310-

14590, and 15730-15930 cm 1 is computed from (3.7) using the line-by-line method, and the

effective mean absorption coefficient k is computed from

k(w) - 1

haT(w) (3.17)

The value of k varies between 0.000135 (cm-atm)sTp 1/2and 0.000155 (cm-atm)sTp 1/2 for a

large range of the scaled 02 amount, w, encountered in the atmosphere. A mean value of

0.000145 (cm-atm)sTp 1/2is used in the broadband radiation model.

The 02 absorption bands are located between water vapor absorption bands. Thus,

water vapor has little effect on the solar heating due to 02. Instead of computing fluxes in the

02 bands, the reduction in fluxes due to 02 absorption is computed from

AF(w) = S(1-e -_47)

The insolation, S, in the spectral regions 7600-8050, 12850-13190,

15730-15930 cm 1 is 6.33% of the total solar flux at the top of the atmosphere.

(3.18)

14310-14590, and

The 02 transmission function, Equation (3.16), is similar to that used by Kiehl and

Yamanouchi (1985), except the pressure scaling of (3.5) is different. It is band-averaged,

does not follow Beer's law, and, hence, cannot be applied to the a-Eddington approximation

for multiple-scattering calculations. We apply (3.16) only to clear atmospheres. For cloudy

atmospheres, a simple scaling approach is used to compute the 02 heating, which is addressed

in Section 6.4.

11
3.8 Absorption due to CO z

The absorption due to CO2 is small but overlaps significantly with the absorption due to

water vapor in certain spectral regions (Figures la and ld). By assuming that CO2 heating in

the stratosphere is not important, Chou (1990) computed the reduction of fluxes due to CO2

based on the pressure scaling of (3.5) and ignoring CO2 absorption in the weak bands at 6400

cm 1 and 7000 cm 1 and in the strong 2300 cm _ band, where the insolation is weak. A

deficiency of this approach is that the heating rate in the upper stratosphere was greatly

underestimated due to the use of the pressure scaling. To demonstrate the importance of the

CO2 heating in the stratosphere, Figure 2 shows the line-by-line calculated heating rate profile

(solid curve) in the spectral region 2200-2400 cm _ and 3500-3760 cm _, where CO2

absorption is strong. The heating rate increases from 1 °C day _at 0.1 hPa to ~ 4.5 °C day 1 at

0.01 hPa. For an atmospheric model which extends to the middle atmosphere, it is desirable

to include this large heating.

13_

rv

-I

to

to

13_

10 2

10 1

1 0 °

1 01

1 02

1 03

' ' ' ' '" "_'

I railtlriiiill ........

.0 0.0 1.0 2.0 3.0 4.0 5.0

Heating Rate (C/day)

Figure 2. The effect of CO 2 absorption in the spectral

regions 2200-2400 cm _ and 3500-3760 cm _ on the

solar heating. Overlapping with water vapor

absorption is included in the calculations of CO2 effect.

The solid curve is line-by-line calculations, and the

dashed curve is the parameterization using the

precomputed table of AN(p c/],to,P) given by

(3.19). The CO2 concentration is set to 350 ppmv, and

the solar zenith angle is set to 60°.

In the 2200-2400 cm _ band, water vapor absorption is negligible; whereas in the 3500-

3760 cm _ band, water vapor absorption is very strong in the troposphere but weak in the

stratosphere, and the effect of CO2 on the tropospheric heating can be neglected. It follows

that the absorption due to CO2 in these spectral regions is not sensitive to water vapor amount,

and an average water vapor path can be used in the parameterization for the flux reduction

due to CO2. By using a fixed water vapor path, the CO2-induced flux reduction at a given

pressure level becomes a function only of the CO2 concentration and the solar zenith angle,

12
AF(Pc/ lu°'P) = Ie-_w(P'[ 1- e-_c(pc/*'°'P'] Svdv (3.19)

where

1 p

G(P) = =-Ikv(P)q(p)dp

/-tg o

(3.20)

P

rc(pc/i.to,p)= Pc yyv(p)dp (3.21)

I'log 0

k and y are the absorption coefficients of water vapor and CO2, Pc is the CO2 concentration

independent of space, q is the water vapor mixing ratio of a typical midlatitude summer

atmosphere, 12o is the cosine of the solar zenith angle, and 12 is taken to be 0.5. Using the

line-by-line method, a table for AF(p Crio,P) normalized by S o is precomputed for

Pc/rio ranging from 250 ppmv to 3300 ppmv and p ranging from 0.01 hPa to 1000 hPa.

The CO2 heating in 2200-2400 cm 1 and 3500-3760 cm 1 derived from the precomputed table

of AF(p C/ rio,P) is shown as the dashed curve of Figure 2. It agrees very well with the lineline-line calculations (solid curve).

For computing the CO2 heating outside the strong bands at 2200-2400 cm _ and 3500-

3760 cm _, the approach of Chou (1990) is followed. Similar to (3.6) and (3.11), fluxes in

the CO2 bands are given by

// m

F(w,u) = f e-(k"_+r"")S, dv = So _2__,2__ _ ni,je , -(kiw+Yiu) (3.22)

i=1 j=l

where w and u are the scaled water vapor and CO2 amounts, and /_.,j is the flux-weighted kdistribution function with the absorption coefficients k i and yj for the two absorbers,

respectively. Equation (3.22) requires (mxn) sets of flux calculations, which becomes

computationally very expensive when used in weather and climate models. Other approaches

than the k-distribution method have to be used for efficient flux calculations when

overlapping of absorption is involved.

The reduction in transmission due to CO2 absorption can be expressed as

AT(w,u) = f e-kv_ (1-e-rv")SvdV / f SvdV (3.23)

13
Since AT(w,u) is a function only of w and u, it is pre-computed using the line-by-line

method for large ranges of w and u encountered in the atmosphere. The integration is over

the total solar spectrum. The absorption due to CO2 in 2200-2400 cm 1 and 3500-3760 cm 1,

is excluded in computing AT(w,u). Fluxes in Band 10 are then simply computed from

F(w,u) = F(w) - SoAT(w,u ) , (3.24)

where F(w) is computed from (3.11).

4. MOLECULAR AND PARTICULATE SCATTERING

4.1 Rayleigh scattering

Rayleigh scattering decreases with decreasing wavenumber according to _V 4. Spectral

values of Rayleigh scattering extinction coefficient are taken from a World Meteorological

Organization report on atmospheric ozone (WMO, 1986). For each of Bands 1-10, a mean

effective Rayleigh scattering coefficient is computed according to the same procedures as in

the computation of the mean effective water vapor absorption coefficient in the PAR region,

(3.13). The Rayleigh scattering coefficients for Bands 1-10 are given in Table 3.

4.2. Cloud and rain single-scattering properties

Radiation routines for atmospheric models usually use only a few broad spectral bands,

with effective optical properties parameterized for each band. The single-scattering coalbedo, 1 - cow varies by several orders of magnitude in the solar spectrum. It is, therefore,

difficult to derive an effective COfor a broad spectral band that can be applied to both strong

and weak absorption situations. There are a number of approaches to deriving CO. Slingo

and Schrecker (1982) derived CO by weighting COy with the solar insolation and linearly

averaging over a spectral band. Espinoza and Harshvardhan (1996) used a square-root

approximation to derive CO. Fu (1996) used a mix of linear and logarithmic averaging

depending on the strength of absorption in a given spectral band. The absorption of solar

radiation by clouds depends not only on COy but also on water vapor and cloud ice/water

amounts. It is clear that no optimal method has been found for deriving CO over a broad

band. It can only be empirically determined based on the amounts of cloud particles and

14
watervaporencounteredin theEarth'satmosphere.

averagingoverabandas

(1- co')= (1- cov)fivSvaV/ fivSvaV

Let us define the linear and logarithmic

(4.1)

log(1 - co") = _ log(1- cov)/3vSvAV / _../3vSvAV (4.2)

where S v

spectral interval where optical properties can be treated as constants.

single-scattering co-albedo of a band is then computed from

is the solar insolation at the top of the atmosphere (TOA), and AV is a narrow

The effective mean

(1 - co) = 7/(1 - co') + (1 - 7/)(1 - co") (4.3)

where r1 is the weight ranging from 0 to 1. The weight r1 should be close to 1 for the

spectral bands with weak absorption and should decrease as absorption increases. Chou et al.

(1997) determined empirically the weight in the various spectral bands that minimize errors

in flux calculations.

Compared to cov, the extinction coefficient (flv) and the asymmetry factor (gv) vary

rather smoothly with v. Their effective mean values over a wide spectral band can be

accurately approximated by

 =Z vSvAV/ZSvAV (4.4)

g = ___gvcovfl, Sv A V / _, covfl, Sv A v (4.5)

Theoretical considerations and radiative transfer calculations have shown that cloud

single-scattering properties are not significantly affected by details of the particle size

distribution and can be adequately parameterized as functions of the effective particle size

(Fu, 1996; Hu and Stamnes, 1993; Tsay et al., 1989). Following Slingo and Schrecker

(1982), we parameterize the single-scattering properties for a broad spectral band by

fl = ao + a 1 / r e

1 - co = bo + blr e + b2re 2

(4.6)

(4.7)

15
g = co + Clre + ¢2re 2 (4.8)

where a, b, and c are regression coefficients, and r e is the effective particle size defined to

be proportional to the ratio of the total volume of cloud particles to the total cross-sectional

area, A C of cloud particles. For spherical water droplets, the effective size is given by

3 C

r. = J r3n(r)dr / J r2n(r)dr - (4.9)

o o 4p. A c

where Pw is the density of water. For hexagonal ice crystals randomly oriented in space, the

effective size is shown by Fu (1996) to be

2_f5 C

ri - ---- (4.10)

3Pi Ac

where Pi is the density of ice, and

volume. The cloud optical thickness, V_, is then given by

r c = flCz

where z is the geometric thickness of a cloud layer.

C is the cloud water/ice mass concentration per unit

(4.11)

The size, shape, and refractive indices are different for ice crystals and water droplets.

The extinction coefficient of ice clouds is smaller than that of water clouds because ice

crystals are much larger than water droplets. The single-scattering albedo and asymmetry

factor of ice clouds and water clouds are also different. Therefore, we parameterize the

single-scattering properties separately for ice and water clouds.

The spectral data (Or, fly, and gv calculated by Fu (1996) for ice clouds and by Tsay

et al. (1989) for water clouds are used to derive C0, fl, and g from (4.1)-(4.5) as functions of

phase and size. By assuming hexagonal ice crystals randomly oriented in space, Fu (1996)

computed the single-scattering parameters of ice clouds using the improved ray-tracing

method of Yang and Liou (1995). A total of 28 size distributions derived from in situ

aircraft measurements were used in the calculations, which included samples from the FIRE

(First ISCCP Regional Experiment) and CEPEX (Central Equatorial Pacific Experiment) field

campaigns. The mean effective size of ice crystals, ri, ranges from 20 ]am to 130 ]am. For

water clouds, the single-scattering parameters of 5 water clouds were computed by Tsay et al.

16
(1989)from the Mie theoryassumingsphericaldropletsanda log-normalsizedistribution.

Theeffectiveradiusof waterdroplets,r_, ranges from 4 _am to 20 _am.

Figures 3-5 show the results of regression using (4.6)-(4.8). The coefficients a, b, and

c are shown in Tables 4-6, respectively. It can be seen in Figure 3 that the extinction

coefficient varies weakly with spectral band but strongly with particle size. For large particles,

it is independent of wavelength for both water and ice clouds. The results shown in Figure 3

for ice clouds are indistinguishable among the four bands. Due to a large particle size, the

extinction coefficient of ice clouds is significantly smaller than that of water clouds. It is

interesting to note that in spite of a large difference in the particle size distribution, the ice

particle extinction coefficients for the 28 clouds nearly fall onto a single curve of (4.6).

Figure 4 shows the single-scattering co-albedo of Bands 9-11. The single-scattering co5

albedo of Band 1-8 is very small (<10) and is set to 0. For a given particle size, the singlescattering co-albedo varies by 3 orders of magnitude among the three IR bands. For a given

spectral band, it is ~ 8 times larger for ice clouds than for water clouds. The asymmetry

factor, shown in Figure 5, varies between 0.78 and 0.94 for different bands and particle sizes.

Since the shape of size distribution has little effect on the single-scattering properties (Hu and

Stamnes, 1993; Slingo and Schrecker, 1982), the results shown in Figures 3-5 should be

representative for a wide range of clouds with various particle size distributions.

For rain drops, a single drop-size distribution is used to compute the spectral

distribution of the single-scattering properties. Fu et al. (1995) used a truncated constantslope gamma function (Manton and Cotton, 1977) with a minimum size of 60 pm, a

maximum size of 1800 pm, and a mean effective size of 162 pm to compute the singlescattering properties. The mean effective extinction coefficient, single-scattering co-albedo,

and asymmetry factor for the 11 spectral bands are shown in Tables 4-6. They are applied to

all other drop-size distributions. The results are provided to us by Prof. Qiang Fu

4.3. Aerosol single-scattering properties

The aerosol optical thickness, single-scattering albedo, and asymmetry factor are not

parameterized. They need to be specified as functions of height and spectral band for input

to the radiation routine.

17
Table 4. Coefficients ao and O1 of the parameterization, Equation (4.6), for the cloud extinction

coefficient. The units of ao and a1 are, respectively, m 2 g 1and m 2 g 1pm.

Spectral Band

Ice Cloud Water Cloud Rain

0 o 01 0 o 01 0 o 01

1-8 3.33e-4 2.52 -6.59e-3 1.65 3.07 e-3 0.0

9 3.33e-4 2.52 -1.01e-2 1.72 3.07 e-3 0.0

10 3.33e-4 2.52 -1.66e-2 1.85 3.07 e-3 0.0

11 3.33e-4 2.52 -3.39e-2 2.16 3.07 e-3 0.0

't2_

%v

O

g

_3

E

LU

0.14 ,, ,i,, ,i , ,,i, ,,i , , ,i , ,,i ,, ,

0.12

0.10

0.08

0.06

0.04

0.02

0.00 '''''''''''''''''''''''''''

20 40 60 80 100 120 140

r i (IJm)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Water Clouds

• Band 1-8

o Band 9

Band 10

.... i .... i .... i .... i ....

5 10 15 20 25

rw (IJm)

Figure 3. The extinction coefficient of ice and water clouds as a function of the effective particle size.

r i and r, are, respectively, the effective size of ice and water particles. The data points are

derived from Equation (4.4), whereas the curves are the regression using Equation (4.6).

18
d

6

o

o

0

m

C_

0000

0000

0000

0000

0000

666

I

66

II

666

I

0

0

o_ m

C_

0000

0000

0000

0000

0000

III

6666

666_

IIII

666_

19
10 _

o

"o

..Q 1 00

6 lO

o

o-_ 1o E

_ 10

O3 10

E 10

10

Ice Clouds

Band 11

Band 10 1

Band 9

10 _

10 °

10 _

10 2

103

10 4

10_

10

0

.... i .... i .... i .... i ....

Water Clouds

Band 11

Band 10

Band 9

. ..,...,...,...,...,..., . . .

.... i .... i .... i .... i ....

20 40 60 80 100 120 140 5 10 15 20 25

r i (#m) r w (pm)

Figure 4. The single-scattering co-albedo of ice and water clouds as a function of the effective particle

size. r i and r, are, respectively, the effective size of ice and water particles. The data points are

derived from Equations. (4.1)-(4.3), whereas the curves are the regression using Equation (4.7).

_3 0.95

u_

0.c

@

E 0.85

E

69

<

0._

0.75

.......... ,oeC,o'ud ......... t

,,, i ,,,i,,, i,,, i,,, i,,, i, , ,

20 40 60 80 100 120 40

r i (pm)

0.95

0.9

0.85

0.8

0.75

.... i .... i .... i .... i ....

Water Clouds

Band 11

d 1-8

l / Band 9

lO

.... i .... i .... i .... i ....

5 10 15 20 25

r w (pm)

Figure 5. The asymmetry factor of ice and water clouds as a function of the effective particle size. r i

and r, are, respectively, the effective size of ice and water particles. The data points are derived

from Equation (4.5), whereas the curves are the regression using Equation (4.8).

2O
5. SURFACE ALBEDO

The surface albedo is an input parameter and is separately specified for the UV and

PAR region (Bands 1-8) and the infrared (Bands 9-11). It is also separately specified for

direct and diffuse fluxes. Therefore, a set of four surface albedos must be specified as input

to the radiation routine.

6. FLUX CALCULATIONS

As mentioned in the previous sections, single values of the gaseous absorption

coefficients and particle single-scattering properties are used for each of the 11 bands, except

for the water vapor absorption coefficient in the IR region (Bands 9-11) where the kdistribution method is used. Therefore, one set of fluxes in the atmosphere and at the surface

are computed for each band in the UV and visible regions (Bands 1-8), and 10 sets of fluxes

corresponding to 10 values of the water vapor absorption coefficient are computed for each

of the IR bands (Bands 9-11). Reflectivities and transmissivities of each atmospheric layer

are first computed using the 5-Eddington approximation (Joseph et al., 1976), fluxes are then

computed by using a two-stream adding method for a composite of layers. The total flux at

each pressure level and at the surface is the weighted sum of these fluxes,

38

F(P) = __, lltiFi(P)

i=1

(6.1)

where l/ti is the fraction of extraterrestrial solar flux contained either in one of the first 8

bands ( S / S o given in Table 3) or in one of the 30 k-intervals in the three IR bands ( h given

in Table 2). The sum of gti is 1.

6.1 Layer optical properties

For each atmospheric layer and spectral band, the effective optical thickness, singlescattering albedo, and asymmetry factor are computed from

7: = Z Vi (6.2)

i

= / y__

i i

(6.3)

= Z giwivi I Z wivi (6.4)

i i

21
wherethesummationis overall gasesandparticles,i. They include ozone, water vapor,

clouds, aerosols, and atmospheric gases (Rayleigh scattering). The single-scattering albedo is

set to 1 for cloud scattering in the UV and visible regions (Bands 1-8) and for Rayleigh

scattering in all bands, whereas it is set to 0 for ozone and water vapor. Cloud ice and water

particles are allowed to mix in a layer. Absorption due to CO2 and 02 is treated differently

and is addressed in Section 6.4

6.2 Layer refleetivities and transmissivities

The transmissivity T and reflectivity R of a layer illuminated by a direct beam are

computed from the 8-Eddington approximation as a function of po and the optical properties

7_, tO, and g defined by (6.2)-(6.4). King and Harshvardhan (1986) have shown that errors

in transmissivity and reflectivity induced by this approximation are -5%. For diffuse

insolation, the transmissivity T and reflectivity R are also computed from the 5-Eddington

approximation but incident angles are approximated by a single value of 53 °,

R - R(_t) (6.5)

T - T(_t) (6.6)

where /2 =cos(53°).

To evaluate the errors induced by this approximation, we use the discrete-ordinate

multiple-scattering algorithm of Stamnes et al. (1988) to compute the transmissivity and

reflectivity of a layer for various incident angles. The diffuse transmissivity and reflectivity

are derived by averaging the direct-beam transmissivity and reflectivity over 2n solid angles,

1

= 2f R(_t)_td_t (6.7)

T = 2f T(/.t)/.td/.t (6.8)

0

where _/ is the cosine of the incident angle depart from the vertical. Errors induced by (6.5)

and (6.6) are shown in Figure 6 for large ranges of the optical thickness and single-scattering

co-albedo. Figure 6 shows that the approximation introduces an error < 0.02 in both the

reflectance and transmittance for wide ranges of the optical thickness and the single-scattering

22
albedo,andanasymmetryfactorof 0.8. For othervaluesof theasymmetryfactor,resultsare

similar.

-1.0-

A

_t -1.5-

I

_-_ -2.0"

o

0

._1 -2.5-

-3.0

o

.0

Reflectonce Difference

-(1.01,5....\

; ;.o.o \

_ 01 001

/ .OlO

i L ' ' i

-0.5 0.0 0.5 110 115 2.0

Tronsmittonce Difference

-1.0

_o -1"5ti

-2.0

0

/ -2.5.

-3£

-- .0

/o °'°

0 O0 0 I

o b b

Ol 01 Oolo 01

-6.5 0.0 0.5 1:0

LOglo-r Log_o"r

;I

o

b

&

\

1.5 2.0

Figure 6. The left panel is the diffuse reflectivity computed from (6.5) minus that computed from (6.7).

The right panel is the diffuse transmissivity computed from (6.6) minus that computed from

(6.8). The cloud asymmetry factor is set to 0.85.

6.3 Two-stream adding approximation and flux calculations

Let us consider a layer divided into an upper sublayer a and a lower sublayer b. The

direct-beam transmissivity T, the direct-beam reflectivity R, the diffuse transmissivity T, and

the diffuse reflectivity R of each of the two sublayers are computed from the 5-Eddington

approximation. By separating the direct and diffuse components of the radiation, the

reflectivity and transmissivity of the layer when illuminated by direct radiation can be

computed from (Chou, 1992)

Rab (po) = Ra(Po)+ -T a{e __°/_°Rb(l.to)+[Ta(Po ) - e-_°/_o ]_b} / (6.9)

(1 - R .R b)

Tab(l_o):e-T'/_°Tb(l_o)+Tb{e-T'/_°-RaRb(l_o)+[Ta(Po)-e-T'/_° ]} ! (6.10)

(1 - R _R b)

23
where T(I.lo), e -r/*'°, and T(I.lo) - e -r/*'° are the total, direct, and diffuse transmissivities,

respectively. The subscripts a and b denote the upper layer and the lower layer.

m

The reflectivity of the layer when illuminated by diffuse radiation from above, R ab, is

generally different from that when illuminated from below, R ab" They can be computed

from

m m

R .b= R .+ T .R bT _,/(1 - R .R b)

R.b= R b+ T bR.T b/(1 - R.R b)

(6.11)

(6.12)

where the overbar and double-overbar denote illumination from above and below the layer,

respectively.

When illuminated by a direct beam radiation at the top of the layer, the reflectivity, R $ ,

and the transmissivity, T $ , at the interface of the sublayers a and b are given by

(6.13)

= + + 1}/ (6.14)

It follows from (6.13) and (6.14) that the upward and downward fluxes at the level

i + 1 / 2 (the lower boundary of the layer i in the atmosphere) can be computed from

Fi+l/2 ?= S{ e-rl'i/t'l° Ri+l,k (]-lo)-]-[Tl,i(]-lo)-e-rl'i/tl° ]R i+l,k }/

(1-Rl,iRi+l,k)

Fi+l/2+= Se-rl,i/tl°-.] - S{e-rl,i/t-lo Rl,iRi+l,k(l.lo)-.]-[Tl,i(l.lo)-e -rl,i/t'l° 1}/

(1-Rl,iRi+l,k)

(6.15)

(6.16)

where S is the insolation at the top of the atmosphere, the index k denotes the surface, and

the subscripts (1, i) and (i+1, k) denote the regions above and below the level i+1/2,

respectively. The transmission T1,i and reflection R1, i of the region above the level are

computed by adding layers from the top of the atmosphere to the layer i using (6.10) and

(6.12) according to the procedures given in Lacis and Hansen (1974). Similarly, the

24
reflectivitiesRi+l. k and R i+l,k are computed by adding layers from the surface up to the

layer i+1 using (6.9) and (6.11).

Fluxes are computed for each of the 8 bands in the UV and PAR regions and for each

of the 30 k-intervals in the IR bands. Total fluxes are the sum of these fluxes given by (6.1).

6.4 Flux reduction due to CO 2 and 02

The reduction of the downward flux due to 02 and CO2 in clear sky conditions, AFZcl,-,

is computed from (3.18), (3.19), and (3.24). To simplify flux calculations in cloudy

conditions, the effect of 02 and CO2 on solar heating is approximated by

zXF*c, , (p) = ZXF*c,r (p),

FFnet _ ,- • ,_s _ .1 cHtp)

aF%(p): c,, L

for P<Pc (6.17)

for p > Pc (6.18)

where F net is the total net downward flux without 02 and CO2 absorption, and Pc is the cloudtop pressure.

7. CLOUD OVERLAPPING

Clouds could occur at various heights with different fractional cover and optical

properties. Nearly all radiative transfer algorithms used in atmospheric models apply only to

a plane-parallel (i.e. horizontally homogeneous) atmosphere. Horizontal inhomogeneity is

not allowed. A straightforward approach to dealing with a partial cloudiness situation is to

divide a region into sections. Within each section, an atmospheric layer is either free of

clouds or filled totally with a homogeneous cloud. Radiative fluxes are then computed for

each section, and the total solar heating is the sum of all sections weighted by their fractional

cover. Depending upon the number of cloud layers and the way these clouds overlap,

computational costs of this approach could be huge. To simplify the computations, a cloud

that partially fills a layer is usually smeared over the entire layer, and the optical thickness, 7:,

is adjusted by a factor dependent on the fractional cloud cover. As demonstrated by a

number of studies (e.g. Harshvardhan and Randall 1985; Cahalan et al., 1994), the effect of

7: on radiation is highly nonlinear, and it is not appropriate to scale 7: (or cloud water path)

linearly by cloud cover. In the NCAR (National Center for Atmospheric Research)

25
communityclimatemodelCCM,the opticalthicknessis scaledby fl.5, where f is the

fractional cloud cover (Briegleb, 1992). In the Goddard GCM, Sud et al. (1993) scaled the

optical thickness by inverting a simple reflection function for clouds when illuminated by

diffuse radiation and assuming no absorption by clouds. As can be expected, the effective

optical thickness corresponding to a cloud smeared to cover the entire layer is a function of

/:, f, and the solar zenith angle, as well as whether scaling is based on reflection or

absorption. Scaling of /: is further complicated by the ways that clouds in various layers

overlap.

7.1. Maximum-random overlapping

Treatments of cloud overlapping in climate studies differ from model to model. In our

radiation parameterization, a combination of maximum and random cloud overlapping

scheme similar to that of Geleyn and Hollingsworth (1979) is used (Chou et al., 1997).

Clouds are identified as high, middle, or low separated roughly by the 400-hPa and 700-hPa

levels. Cloud layers close to each other are likely to be related; therefore, cloud layers in each

of the three height groups are assumed to be maximally overlapped (left panel of Figure 7),

whereas overlapping among the three groups is assumed to be random. Scaling of /: is

applied only to the maximally-overlapped clouds within each of the three height groups, but

not to the randomly-overlapped clouds among different height groups (right panel of Figure

7). Within each height group, clouds are smeared over the extent of the maximum cloud

amount, fro, of that height group by scaling /: (right panel of Figure 7). By assuming

random overlapping of clouds among different height groups, the atmosphere is then divided

into 2" sections, where n _<3 is the number of height groups containing clouds (right panel

of Figure 7). Within each section, a layer is either totally cloud-filled or cloud-flee. Fluxes

are first computed for each section and then summed over all sections weighted by the

fractional cover of the individual sections.

As examples, the fractional cover of the clear section, Ac/r, is computed from

A ,r= (1- L,,O(1- -

(7.1)

26
wherethe subscripts1, 2, and 3 denotesthe upper, middle, and lower cloud groups,

respectively.Similarly,thefractionalcoverof thesectionwithonly high cloudsis computed

from

A =f,,,l(1-f,,,z)(1-f,,,3) (7.2)

Cloud Overlapping

I I ,

I I

_'_'_ i

I I

I I

' V//////////_ '

,i K_,,\\\\\\,,,, S ,i

I I

I I

, _\\\\\\\\\_ ,

I I

I I

Surface

400 hPa

700 hPa

Optical-thickness Scaling

_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_iii!

Surface

Figure 7. The maximum-random cloud-overlapping scheme. Clouds are identified as high, middle, and low

separated roughly by the 400 hPa and 700 hPa levels. Clouds are assumed maximally overlapped

within each height group and randomly overlapped among different height groups (left panel).

Scaling of the optical thickness applies only to the maximally-overlapped cloud layers (right panel).

27
7.2. Scaling of optical thickness

Cloud optical-thickness scaling used to "smear" the clouds to fm within each height

group is based on the maximum-random cloud overlapping assumption shown in Figure 7.

To simplify the scaling of 7r, we assume that reflection is more important to climate than

absorption, since the former affects the total heating of the earth-atmosphere system, but the

latter only redistributes the heating within the atmosphere, as well as between the atmosphere

and the surface. As stated in Sections 6.2 and 6.3, our radiation algorithm requires

calculations of reflection/transmission of a cloud layer separately for direct and diffuse

radiation. By smearing cloud cover over the extent of the maximum cloud cover of

respective height-group, the optical thickness of a cloud layer is scaled by factors Z_ and Z_,

rb = Z, r for direct radiation (7.3)

7rf = Z Tr for diffuse radiation (7.4)

so that albedos of the layer remain the same before and after scaling,

R( Zb,r e,Po ) = f' R( Z, r e,Po) for direct radiation

-R(rf,r_) = f '-R(_,re) for diffuse radiation

(7.5)

(7.6)

where Po is the consine of the solar zenith angle, R is the albedo averaged over the entire

solar spectrum for direct incident radiation, R is the albedo for diffuse incident radiation

given by

1

R(r, re) = 2IR(r, re,lJ)lJd p (7.7)

f is the normalized cloud cover given by

f = f / f,_ (7.8)

and p is the cosine of the zenith angle of an incident beam. Using the scaling of (7.3) and

(7.4), a layer with a fractional cloud cover f and an optical thickness z is reduced to a layer

with a fractional cloud cover f_ and an equivalent optical thickness /:b for direct radiation

and /:f for diffuse radiation. It is noted that the scaling is a function of the incident angle,

and that optical thickness is scaled separately for direct radiation and diffuse radiation.

28
Usingvaluesof r e representative of the water and ice cloud samples used in Section 4.2,

(,o and g are computed from (6.3) and (6.4) for the individual bands given in Table 1.

Cloud albedos R and R averaged over the solar spectrum are computed using the discreteordinate multiple-scattering algorithm of Stamnes et al. (1988). The scaling functions Z_

and Z_ are then derived from (7.3)-(7.6) by giving f', R(v, re,po)and R(v, re). Sample

calculations show that Z_ and _'_ depend only weakly on re and the phase of cloud

particles. Therefore, they are reduced to z_(f',v,]lo) and z_(f',v) by taking the mean

values of the scaling functions for various particle sizes and phases.

The above scaling ensures that the reflection of a cloud layer is the same before and

after the scaling. However, errors are expected in the calculations of atmospheric and surface

heating when there are other cloud layers present. In addition to the solar zenith angle, cloud

cover, and optical thickness, an optimal cloud scaling should, in principle, also depend on the

fractional cover and optical thickness of other cloud layers. This situation makes scaling of

the optical thickness extremely difficult. To simplify the problem, we made simple

adjustments to the scaling functions, Zs and Zs, so that errors in the fluxes at the TOA and

the surface are minimized for the simplest cloud situation, which has two cloud layers in only

one of the three height-groups. For the two-cloud situation, scaling is applied to the cloud

with a smaller cloud cover (see right panel of Figure 7). It is noted that scaling is not

required when there is only one cloud layer in a given height-group. The scaling functions

are adjusted according to

Z = Zs + (f -Zs) (_ (7.9)

X = Xs + (f -Xs )(_, (7.10)

where _ and _ are constants empirically (trial and error) chosen to minimize errors in the

fluxes at the TOA and at the surface. The errors are defined as the differences between the

flux computed with the scaling of V and the flux computed with the sky divided into

horizontally homogeneous sections (three sections in the two-cloud situation). The adjusted

scaling functions preserve the shapes of Zs and Zs" The results for the scaling functions are

saved as tables for later use. It is found that optimal values of _ and _ are 0.2 and 0.3,

respectively. The results for the scaling functions Z and Z are shown in Figure 8. It can be

29
seenthat z(f', 7:,1-1o)< f', and Z --->f' as 71--->0.

are not sensitive to 7r.

For an optically thick cloud, Z and Z

The scaling functions derived for the two-cloud situation is applied universally to all

situations even if there are more than two cloud layers in a given height group. It implicitly

assumes that, in a given height group, accurate scaling of lr is not critical if there are two

cloud layers above.

With the sky divided into 2" sections, fluxes are computed for each section. Instead of

(6.1), the total flux is computed from

F(p) = _ AjF_,j (p)

/=1 '=

(7.11)

1.0

0.8

0.6

0.4

0.2

0.0

0.0

' ;o'=&..........

1 • ....'" ,t"

.......

0.2 0.4 0.6 0.8 1.0

Cloud Cover, f'

1.0

0.8

0.6

0.4

0.2

0.0

O.

.-_ ..'64

..,- .s"

i , , , i , , ,

0.2 0.4 0.6

, , , i , , , i , , • i , , , i"_t

__j_I_I_i

i_i_ _

0.8 .0

Cloud Cover, f'

Figure 8. Scaling functions for the cloud optical-thickness, Z and Z, for direct (shown only for Po =

0.3) and diffuse radiation, respectively. The value of f' is the normalized cloud cover defined

by (7.8). Numbers in the figure are the cloud optical thickness. The dots are Z = f' 1.5used in

the NCAR CCM model (Briegleb, 1992).

where N = 2", and Aj is the fractional cover of the jth section. The sums of Aj's and

_i's are both 1. It needs to compute the reflectivities and transmissivities of each of the

3O
atmosphericlayersonly oncebutnotN times. Dependinguponthenumberof sectionsN,

the total computing time is about three times larger than the computing time for a single

section.

8. SAMPLE CALCULATIONS

8.1. Effect of absorption and scattering in minor bands

Concerned with the huge time required for computing radiative transfer in weather and

climate models, solar heating in minor absorption bands has often been neglected. The

individual contributions of these minor bands to the atmospheric heating is small, but

collectively they are not negligible (Chou, 1999). The minor absorption and scattering which

are often neglected include the absorption due to water vapor in the photosynthetically active

radiation (PAR) spectral region (Band 8) and in the thermal infrared (1000-2600 cm 1), the

ozone absorption and Rayleigh scattering in the near infrared (Band 9), and 02 and CO2

absorption in wide spectral ranges. Ramaswamy and Freidenreich (1991) have shown the

importance of the two minor water vapor regions. Table 7 shows the effects of these minor

absorption and scattering bands on the surface solar heating for the typical midlatitude

summer atmosphere with a CO2 concentration of 350 ppmv, a solar zenith angle of 60 °, and a

surface albedo of 0. Using a spectral resolution of 0.002 cm 1 for water vapor, 02, and CO2,

and 0.003 pm for 03, the total effect of these minor bands is computed with the discreteordinary scattering algorithm. The surface solar heating is reduced by 17 W m 2. The

atmospheric solar heating increases by a similar amount, ~15 W m 2 (not shown) when these

minor bands are included. This amounts to about 10% of the total atmospheric heating,

which is not negligible in the solar heating calculations. The results of the parameterizations

for these minor bands are also shown in the table, which are accurate compared to the detailed

calculations.

8.2. Clear-sky fluxes and heating rates

Clear-sky fluxes and heating rates computed with the parameterization are validated

with high spectral-resolution calculations. The results are shown in Table 8 for fluxes and

Figure 9 for the heating rate. Calculations are for the mid-latitude summer atmosphere with a

31
Table 7. Effects of minor absorption and scattering on the downward surface solar flux from detailed

high spectral-resolution calculations and parameterizations. Fluxes are computed for a typical

mid-latitude summer atmosphere. The CO2 concentration is set to 350 ppmv. The solar

zenith angle and the surface albedo are set to 60 ° and zero, respectively. Units are W m 2.

Detailed Calculations Parameterization

Water vapor in PAR

Ozone in near infrared*

Oxygen in near infrared*

CO2 in middle infrared*

Rayleigh in near infrared*

H20 in thermal infrared**

Total

-2.53 -2.42

-0.56 -0.54

-4.29 -4.18

-3.30 -3.57

-3.12 -2.82

-3.09 * * *

-16.89

Overlapping with water vapor absorption is included.

1000 cm 1<v<2600 cm 1.

The effect cannot be assessed because a separate parameterization excluding

the thermal infrared is not done.

CO2 concentration of 350 ppmv. The solar zenith angle is set to 60 °, and the surface albedo is

set to 0.2. For the high resolution calculations, the spectral interval is chosen to be 0.003 pm

for 03 and 0.002 cm 1 for water vapor, 02, and CO2. Only the absorption due to water vapor,

03, 02, and CO2 are included. Scattering due to clouds, aerosols, and gases are excluded.

Absorption of solar radiation in the atmosphere (ATM) and net downward fluxes at the

top of the atmosphere (TOA) and the surface (SFC) are shown in Table 8 for different bands

and the total spectrum. The parameterization introduces an error of <1% in the TOA and

surface fluxes and the atmospheric heating. Figure 9 shows the heating rate as functions of

the logarithm of pressure (left panel) and of pressure (right panel). The parameterization

computes very accurately the heating rate below the 1-hPa level but with an error of 5% of

the peak heating above. Most of the error is attributable to the use of the pressure scaling in

the calculations of absorption due to water vapor, 02, and CO2.

32
Table 8. Net downward solar fluxes at the top of the atmosphere (TOA) and at the surface (SFC) and the

radiation absorbed in the atmosphere (ATM) computed using high spectral resolution calculations

and the parameterization. The atmosphere used is typical of a mid-latitude summer with a CO2

concentration of 350 ppmv. The solar zenith angle and the surface albedo are set to 60 ° and 0.2,

respectively. Calculations include the absorption due to water vapor, 03, CO2 and 02 covering the

spectral range from 0.175 pm to 10 pm. Scattering is excluded. Units are W m 2.

High Spectral-Resolution Parameterization

TOA SFC ATM TOA SFC ATM

Band 1-7 47.7 33.7 14.0 46.7 33.0 13.7

Band 8 217.7 203.2 14.5 218.0 203.9 14.1

Band 9 186.7 137.8 48.9 186.5 134.9 51.6

Band 10 101.5 51.0 50.5 100.3 51.7 48.6

Band 11 28.0 7.8 20.2 27.6 8.2 19.4

Total 581.5 433.3 148.2 579.1 431.7 147.4

1 0

1 0

13..

1 0 _

_ 1 0 _

1 02

1 02

' (}, ......

L'"-_al Resolution.

Paramete_

_, , , , .... , .... , .... , ....

5 10 15 20

Heating Rate (C/day)

0

200

400

600

800

1000

5 0.0

%:iymeteriz tion

High Spectral Resolution '._

0.5 1.0 1.5

Heating Rate (C/day)

Figure 9. The heating rate profiles of a mid-latitude summer atmosphere computed with the line-byline method (solid curves) and the parameterization (dashed curves). The heating is expressed as

a function of the logarithm of pressure in the left panel and as a function of pressure in the right

panel. The solar zenith angle is set to 60 °, surface albedo is set to 0.2, and the CO 2

concentration is set to 350 ppmv. Absorption due to water vapor, 03, 02, and CO 2 are included.

Scattering is excluded.

33
8.3. Cloudy-sky fluxes and heating rates

Fluxes and heating rates in a cloudy atmosphere derived from the parameterization and

the detailed absorption and scattering calculations are compared. A deck of stratus cloud

covering the entire sky is inserted between 800 hPa and 920 hPa. The effective radius of

liquid cloud particles, r,, is set to 12 gm, and the optical thickness in the 0.55 ]am visible

region, /:,is, is set to 9.7. Based on results from Mie calculations for those r_ and /:,is, the

extinction coefficient in the visible region, [3_is, is 0.1303 g m 2, and the total cloud water

amount, Cz given by (4.11), is 74.4 g m 2. The cloud layer is divided into 5 sublayers with a

thickness of 24 hPa. For each sublayer, the cloud water amount is 14.9 g m 2, and the optical

thickness in the visible region is 1.94.

For the detailed calculations, the line-by-line method and the discrete-ordinate

scattering algorithm are used. The spectral distributions of the cloud extinction coefficient,

fly, the single-scattering albedo, C0v, and the asymmetry factor, gv, for r,=12 gm are

derived from Mie calculations. The spectral distribution of the optical thickness is scaled

according to

For the parameterization, the band-averaged cloud extinction coefficient, the single-scattering

albedo, and the asymmetry factor are derived from (4.6)-(4.8), and the optical thickness is

derived from (4.11). Fluxes are computed using the two-stream adding approximation given

in Section 6.3.

Table 9 shows the net downward fluxes at the TOA and surface and the solar heating of

the mid-latitude summer atmosphere using the detailed calculations and the parameterization.

The solar zenith angle and the surface albedo are set to 60 ° and 0.2, respectively. The

discrepancy between the two sets of calculations is very small in the surface flux and is ~2%

in the total TOA flux. Figure 10 shows the heating rate profiles of the two sets of

calculations. They agree very well both inside and outside the cloud.

34
Table 9. Same as Table 8 except scattering by gases and cloud is included. A deck of stratus cloud

with an optical thickness of 9.7 in the visible region is located between 800 and 920 hPa.

Units are W m 2.

Detailed Calculations Parameterization

TOA SFC ATM TOA SFC ATM

Band 1-8 138.7 105.8 32.9 137.8 107.3 30.5

Band 9 110.6 62.9 47.7 106.4 61.9 44.6

Band 10 78.5 17.9 60.6 74.1 18.5 55.7

Band 11 27.1 0.7 26.4 28.0 0.1 27.9

Total 354.9 187.2 167.7 346.4 186.9 159.6

13._

r"

v

13._

200

400

600

800

1ooo

' iiOisirl ............

...... Parameterization

_ Cloud Layer:800-920 hPa

..........

1 2 3 4 5 6 7 8

Heating Rate (C/day)

Figure 10. Same as Figure 9, except a

cloud deck is inserted between 800 and

920 hPa. The cloud optical thickness is

9.7 in the visible region, and the

effective radius of cloud liquid drops is

12 pm.

ACKNOWLEDGMENTS

This work was supported by the Global Atmospheric Modeling and Analysis Program,

NASA/Office of Earth Science.

35
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Yang, P., and K.-N. Liou, 1995: Light scattering by hexagonal ice crystals: Comparison of

FDTD and geometric optics models. J. Opt. Soc. Amer., 12, 162-176.

38
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

June 1999 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Technical Report Series on Global Modeling and Data Assimilation

Volume 15--A Solar Radiation Parameterization for Atmospheric Studies Code 913

6. AUTHOR(S)

Ming-Dah Chou and Max J. Suarez

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS (ES)

Climate and Radiation Branch

Goddard Space Flight Center

Greenbelt, Maryland 20771

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS (ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PEFORMING ORGANIZATION

REPORT NUMBER

99A01083

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

TM-1999-104606, Vol. 15

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION / AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category: 46

Report available from the NASA Center for AeroSpace Information,

7121 Standard Drive, Hanover, MD 21076-1320. (301) 621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The solar radiation parameterization (CLIRAD-SW) developed at the Goddard Climate and Radiation Branch

for application to atmospheric models are described. It includes the absorption by water vapor, 03, 02, CO2,

clouds, and aerosols and the scattering by clouds, aerosols, and gases. Depending upon the nature of absorption,

different approaches are applied to different absorbers. In the ultraviolet and visible regions, the spectrum is

divided into 8 bands, and single 03 absorption coefficient and Rayleigh scattering coefficient are used for each

band. In the infrared, the spectrum is divided into 3 bands, and the k-distribution method is applied for water

vapor absorption. The flux reduction due to 02 is derived from a simple function, while the flux reduction due

to CO2 is derived from precomputed tables. Cloud single-scattering properties are parameterized, separately

for liquid drops and ice, as functions of water amount and effective particle size. A maximum-random

approximation is adopted for the overlapping of clouds at different heights. Fluxes are computed using the

Delta-Eddington approximation.

14. SUBJECT TERMS

Solar Radiation, Atmospheric Models, k-Distribution Method,

Delta-Eddington Approximation, Minor Absorption

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

4O

16. PRICE CODE

20. LIMITATION OF ABSTRACT

LTL

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

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