NASA/TM-2001-104606, Vol. 19

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Goddard Space Flight Center, Greenbelt, Maryland

Volume 19

A Thermal Infrared Radiation

Parameterization for Atmospheric Studies

Ming-Dah Chou and Max J. Suarez

NASA Goddard Space Flight Center, Greenbelt, Maryland

Xin-Zhong Liang

University of lllinois at Urbana-Champaign, Champaign, Illionois

Michael M.-H. Yah

Science Systems and Applications, Inc., Lanham, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

July 2001
NASA Center for AeroSpace Information

7121 Standard Drive

Hanover, MD 21076--1320

Price Code: A 17

Available from:

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

Price Code: A10
ABSTRACT

This technical memorandum documents the longwave radiation parameterization developed

at the Climate and Radiation Branch, NASA/Goddard Space Flight Center, for a wide variety of

weather and climate applications. Based on the 1996-version of the Air Force Geophysical

Laboratory H/TRAN data base (Rothman et al., 1998), the parameterization includes the

absorption due to major gaseous absorption (water vapor, C02, 03) and most of the minor trace

gases (N20, CH 4, CFC's), as well as clouds and aerosols. The thermal infrared spectrum is

divided into nine bands. To achieve a high degree of accuracy and speed, various approaches of

computing the transmission function are applied to different spectral bands and gases. The

gaseous transmission function is computed either using the k-distribution method or the table

look-up method. To include the effect of scattering due to clouds and aerosols, the optical

thickness is scaled by the single-scattering albedo and asymmetry factor. The optical thickness,

the single-scattering albedo, and the asymmetry factor of clouds are parameterized as functions of

the ice and water content and the particle size. The parameterization can accurately compute

fluxes to within 1% of the high spectral-resolution line-by-line calculations. The cooling rate can

be accurately computed in the region extending from the surface to the O.Ol-hPa level.

The computer code for this longwave parameterization is very easy to use. It has been

implemented in atmospheric models for cloud, weather, and climate studies at many government

institutes and universities. The code and sample calculations are accessible at

http://climate.gsfc.nasa.gov/-chou/clirad_lw.

.,.

111
iv
TABLE OF CONTENT

1. INTRODUCTION .............................................................................. (1)

2. INFRARED TRANSFER EQUATIONS ...................................................... (2)

3. SPECTRAL BANDS ............................................................................. (4)

3.1 The 10 Bands .................................................................................... (4)

3.2 Planck-Weighted Band Integrals .............................................................. (4)

3.3 Band-Integrated Planck Functions ...................................................... (6)

3.4 Sensitivio, of Upward Flux to Surface Temperature ....................................... (7)

4. BAND-AVERAGED TRANSMISSION FUNCTIONS OF GASES ..................... (8)

4.1 The k-distribution method .................................................................... (8)

4.2 Pre-computed Transmittance Tables ...................................................... (15)

4.3 One-parameter scaling for water vapor continuum absorption ........................ (18)

4.4 Overlapping of Gaseous Absorptions ................................................ (19)

4.5 Specail Treatment of the 15-pm Bamt ................................................... (20)

5. SPECIAL TREATMENT OF THE 17-/Jm N20 BAND ................................... (23)

6. CLOUDS ............................................................................................. (24)

6.1 Cloud Single-Scattering Properties ............................................................ (24)

6.2 Layer optical properties ..................................................................... (29)

6.3 Paramaterization for Cloud Scattering ....................................................... (29)

6.4 Cloud Overlapping ............................................................................ (30)

7. AEROSOLS ......................................................................................... (33)

8. VERTICAL DISCRETIZATION

8.1 Downward and Upward Fluxes

8.2 Treatment of Adjacent Layer

............................................................. (34)

............................................................... (34)

.................................................................. (37)

8.3 The k-Distribution Transmission Functions ............................................... (39)

8.4 Table Look-Up for Transmission Functions ............................................... (40)

9. SUB-GRID VARIATIONS OF LAND SURFACE ......................................... (41)

10. COMPARISONS WITH LINE-BY-LINE CALCULATIONS ........................... (44)

V
! 1. CONCLUDING REMARKS ................................................................... (50)

ACKNOWLEDGMENT ............................................................................... (51 )

REFERENCES .......................................................................................... (52)

vi
List of Figures

1. Contribution from a differential layer at p' to the downward radiance at p. .......... (2)

. Schematic of the absorption coefficient for CO2 and for water vapor line and continuum

absorption in the 15-pro region (Band 3). The subdivision of the band into three sub-bands is

made to accurately account for the large variations of the absorption coefficients.

...................................................................................................... (21)

. Schematic showing two groups of randomly overlapped clouds. Cloud layers within each

group are assumed maximally overlapped. CU and CL are the clear line-of-sight of the upper

group and lower group, respectively. The clear line-of-sight between the levels p and p' is

CUCL as shown in (6.21) ................................................................. (33)

4. The vertical grid and placement of various quantities for an atmosphere consisting of L layers.

Quantities defined at the layers, such as the Pianck Flux, B are denoted by half-integer

subscripts, and quantities defined at the levels separating them, such as the downward and

upward fluxes F, by integer subscripts. The transmittance shown is for a multi-layer region

bounded by levels I and l'. Note that the surface is treated as a fictitious layer at L+3/2

............................................................................ (35)

5. The cooling rate computed for a clear mid-latitude summer atmosphere using the line-by line

method (solid curves) and the "HIGH" option of the parameterization (dashed curves).

Cooling is due to water vapor molecular line and continuum absorption, as well as C02 and

03 absorption .......................................................................... (45)

6. Same as Figure 5 except for a clear sub-arctic winter atmosphere ......................... (46)

vii
viii
List of Tables

1. Spectral bands, absorbers, and transmittance parameterizations ........................... (5)

2. Coefficients for computing the spectrally integrated Planck flux from Equation (3.11). Units

of 0 in (3.1 I) are Kelvin ........................................................................ (7)

3. Spectral bands, reference pressure and temperature, and the coefficients for the pressure and

tcmperaturc scaling used in the k-distribution method. Band 3 is divided into three subbands ............................................................................................... (9)

4. Paran_'tcrs for the transmittance given by (4.10) due to water vapor line absorption, k I /

is the first absorption coefficient, and rl is the constant given in (4.12). Ag is the kdistribution function (4.11). Band 3 is treated separately in Section 4.5. Units of k are g J

cm: ............................................................................................... (12)

5. Parameters for computing theflux transmission functions due to N20 and CH4 using the kdistribution method. (pr = 500 hPa and 0_=250K) ........................................... (14)

6. Same as Table 5, except for the absorption due to CO, .................................... (14)

7. Same as Table 5, except for the minor CFC absorption bands. There is only one k-interval for

each CFC band, and there is no pressure scaling applied, i.e. m=O ....................... (15)

8. The first values, lOglo(We_)l and logio(pe_b, the intervals, Aloglo(W_) and AlogJpetr), and the

numbers, nw and np, of loglo(W_ H) and logm(p,_) of the tables a, b and c for transmittance

-2

calculations using (4.16). Units of %it are gcm for water vapor and (cm-atm)srp for CO2 and

03, and the unit of p_ is hPa ................................................................ (17)

9. The effective absorption coefficient k c / -_ for the water vapor continuum absorption for

Bands 3-7. Units of k c / "_ are g-_ crn 2................................................... (19)

10. The first value of the absorption coefficient k 1 / _. for the water vapor line absorption, the

value of rl that k, = rlk,._, and the Planck-weighted k-distribution function (A_) given by

(4.29) and (4.30) of the three sub-bands in the 15/zm region ............................. (22)

ix
11a. Coefficients for computing the extinction coefficient for cloud ice panicles from (6.4a).

Units of the particle size in (6.4a) are tam................................................... (26)

i lb. Coefficients for computing the extinction coefficient for cloud liquid water droplets from

(6.4b). Units of the particle size in (6.4b) are lam ......................................... (26)

12a. Coefficients for computing the single-scattering albedo for cloud ice panicles from (6.5).

Units of the particle size in (6.5) are tam.................................................... (27)

12b. Coefficients for computing the single-scattering albedo for cloud liquid water droplets from

(6.5). Units of the particle size in (6.5) are pm ........................................... (27)

13a. Coefficients for computing the asymmetry factor for cloud ice particles from (6.6). Units of

the particle size in (6.6) are _tm............................................................. (28)

13b. Coefficients for computing the asymmetry factor for cloud liquid water droplets from (6.6).

Units of the particle size in (6.6) are lam................................................... (28)

14. Downward fluxes at the surface, F+_,., and upward fluxes at the top, F+,op,for a clear midlatitude summer atmosphere computed using a line-by-line method and the parameterization

with the HIGH option. Units of the fluxes are Wm 2.................................... (47)

15. Same as Table 14, except for a clear sub-arctic winter atmosphere ..................... (48)

16. Effects of the minor absorption bands on the fluxes at the top of the atmosphere AFr,op and

the surface Ab-4_cas calculated using the LW radiation parameterization. See the text for the

gas concentrations used in flux calculations. Units of the fluxes are Wm-2..... (49)

X
I. _TRODUCTION

Thermal infrared (or Iongwave, LW) radiation plays a crucial role in affecting weather,

climate, and the sensitivity of climate to external radiative forcing. It is, therefore, important to

have an accurate LW radiation parameterization in atmospheric models for weather and climate

studies. In numerical model simulations of weather and climate, calculations of LW fluxes can

easily take a third or more of the total computing time. As the spatial and temporal resolution of

the models increases and the treatment of physical processes improves, it becomes clear that we

need a fast and accurate LW radiation parameterization.

Detailed calculation of the LW fluxes involves three sets of integration: spectral integration,

vertical integration, and directional integration. The spectral integration is the most time

consuming, but it is the vertical integration that makes the parameterization for the LW radiation

particularly difficult. If the atmosphere were vertically homogeneous in pressure and

temperature, then the absorption coefficient would be a function only of wavenumber, and

wavenumbers with the same absorption coefficient would be radiatively identical. The spectral

integration could then be greatly simplified by using the k-distribution method (c.f Arking and

Grossman, 1972), in which only one set of radiative transfer calculations is needed for all the

wavenumbers with the same absorption coefficient. In the real atmosphere, however, the

dependence of the absorption coefficient on pressure and temperature varies with wavenumber,

and in principle no two spectral intervals can be treated identically in the radiative transfer

calculations. The difficulty is augmented by the narrowness of molecular absorption lines, which

makes the absorption coefficient vary rapidly with wavenumber. Thus, it requires a very high

spectral resolution to obtain spectrally integrated fluxes with high accuracy. To further

complicate the flux calculations, there are numerous absorbers that have to be taken into account.

Except for water vapor, those absorbers spread over the LW spectrum in narrow bands. The

effects of those bands need to be computed individually.

At the Goddard Climate and Radiation Branch, we have developed various LW radiation

parameterizations for the major water vapor, C02 and 03 absorption bands (Chou and Kouvaris,

1991; Chou et al., 1993). These IR radiation parameterizations have been shown to be accurate

and efficient in computing cooling rate not only in the troposphere and lower stratosphere, but

also in the middle atmosphere (up to the O.Ol-hPa level). The code was documented in a NASA

Technical Memoramtum (Chou and Suarez, 1994).
This technical memorandum is an extension of Chou and Suarez (1994) by including the

absorption due to minor trace gases of N20, CH4, and CFC's (Kratz et al., 1998) and

parameterizations for the absorption and scattering due to clouds (Chou et al., 1999). Aerosols are

also included, but with optical properties specified as input parameters. This LW radiation code

has been implemented in the Goddard general circulation models (Bacmeister and Suarez,, 2001;

Sud and Mocko, 1999) and the cloud ensemble model (Tao et al., 2001). It has also been used at

various universities and government institutes.

2. INFRARED TRANSFER EQUATIONS

Let us ct_nsider a thin atmospheric layer at pressure p' with a temperature 0', which has a

differential pressure thickness dp' and an optical thickness dxv at wavenumber v (Figure 1). The

radiance emitted by this layer is Rv(O')dxv/p, where/a is the cosine of the angle between the beam

and the vertical, and R,. is the Planck function. Let us further consider a lower level at pressure p

at which we wish to compute the downwelling flux, and let us assume that between p and p'

there is a mixture of gaseous, cloud, and aerosol absorbers with monochromatic optical thickness

x_(p,p'). The transmittance between p' and p in the direction/.t is e -x_(p'p')/_t. When scattering

is neglected, the contribution of the layer between p' and p' +dp' to the radiance at the level p

is then given by

d/v'['(_t) = Rv (0')d-'_- e-XV (p,p')la (2.1)

dL, (_)i¢

P

Figure 1. Contribution from a differential layer at p' to the downward radiance at p.
The contribution of the layer to the downward flux at p can be derived by angular

integration of (2.1) over the hemisphere,

dFv $ = 2_ Ii dlv $ ( g)gd_

= 2rc/?v (0 ,)fl d_v e_, v (P'P')IggdP - (2.2)

do g

= -Re (0') dTv (p, p')

where By(0)= xRv(0) is the Planck flux, and Tv (p,p') is the flux transmittance for isotropic

radiation given by

Tv(p,p,)= 2 e-%(P,P')Ogdp. (2.3)

The total downward flux integrated over heights and wavenumbers is then given by

Op' dp' (2.4)

Similarly, the upward flux can be expressed as

(Os ' (,,,,,,)

L "Ps rp

where V-.vis the surface emissivity, p, is the surface pressure, and 0, is the Earth's surface skin

temperature. Finally, the cooling rate is given by

_t Cpg

where cp is the heat capacity of air at constant pressure, and g is the gravitational acceleration.
3. SPECTRAL BANDS

3.1 The 10 Bands

For computing thermal IR fluxes, the spectrum is divided into 9 bands and a subband.

Table 1 shows the spectral ranges for these 10 bands, together with the absorbers involved in each

band. The water vapor line absorption covers the entire IR spectrum, while the water vapor

-1

continuum absorption is included in the 540-1380 cm spectral region. The absorption due to C02

-1

is included in the 540-800 cm region, and the absorption due to Os is included in the 980-1100

-1

cm region. The minor absorption due to CH4, N20, CFC's, and C02 is scattered between 800

-1 -/

cm and 1380 cm region in Bands 4-7. The absorption due to N20 in the 17-wn region is

included in Sub-band 3a and is identified as Band 10.

3.2 Pianck- Weighted Band Integrals

From (2.4) and (2.5), the downward and upward fluxes integrated over a spectral band i can

be written as

Fi "1"(p)= Bi(e' ( _ (P,,P )

_k _P' i dp' (3.1)

FiT (p)=13iBi(Os)Ti(p, ps)+ fP t}(O')(OT"(P; P' )) dp '

Ps _ Op i

(3.2)

+ (1 - e i )Fi $ (ps)Ti(p, ps )

where

F,(p) = Ia Fv(p)dv (3.3)

vi

Bi(O ) = Ia By (0)dr (3.4)

Vi

1 Ia Bv (0')1" v (p, ff )dr (3.5) T/(p,p' )= Bi(O' ) vi

4
Table1. Spectral bands,absorbers, andtransmittanceparameterizations.

Band SpectralRange Absorber Optionsfor

(cm-1) Transmittance Parameterization

"LOW....HIGH"

1 0-340

2 340-540

3a 540-620

3b 620-720

3c 720-800

4 800-980

5 980-1100

6 1100-1215

7 1215-1380

8 1380-1900

9 1900-3000

10 540-620

He0 line K T

H20 line K T

HeO line K T

H20 continuum C C

C02 K T

HeO line K K

HeO continuum C C

C02, Fll, F12, F22 K K

HeO line, COe, FI 1 K K

H20 continuum C C

03 T T

H20 line K K

He0 continuum C C

n20, CH4, F12, F22 K K

HeO line, N20, CH4 K K

He0 continuum C C

HeO line K T

He0 line K K

H20 line, COe, N20 K K

HeO continuum C C

K:

T:

C:

k-distribution method with linear pressure scaling.

Table look-up with temperature and pressure scaling.

One-parameter temperature scaling for water vapor continuum absorption.
(OT(p,p'))- 1 ;/ Bv(O,)OTv(p,p')d v

ap' i 8i(o') v, ag

are the Planck-weighted band integrals, and Av_ is the width of the i'hband.

(3.6)

Within each spectral band, either the range of By is sufficiently small or the shape of By is

sufficiently independent of temperature that we can make the following approximation

1

;a Bv(O°)Tv(p'ff )dr (3.7) Ti(P'P )= Bi_Oo)' v,

Op i = ni(Oo ) By ,- o, Off _ff

i

where 0o is a typical value of the atmospheric temperatures, which is set to be 250K.

With the approximation (3.7) and (3.8), the band-integrated fluxes reduce to

=_o' , _ar_(p,p' Fi$ (p) Bi(O )dp' (3.9)

- _p,

fP OTi(P, p

F/t (p) =e.iBi(Os)Ti(p, ps)+ t_(O') ') dp'

Ps Off (3.10)

+ (1 - ei)Fi $ (Ps)Ti(p, ps )

3.3 Band-Integrated Planek Functions

The spectrally integrated Planck fluxes were pre-computed for each band and then fit by a

yh-degree polynomial in temperature,

5

Bi(0) = c/,0 +ECi, nOn (3.11)

n=!

When integrated over all bands, errors in this regression are negligible (< 0.1%) for 150K < 0 <

350K. The coefficients c's are listed in Table 2.

6
Table 2. Coefficients for computing the spectrally integrated Planck flux from Equation

(3.11). Units of 0 in (3.11) are Kelvin.

Band Co cl c2 C3 C4 C5

1 5.344e+0 -2.062e- l 2.533e-3 -6.863e-6 1.012e-8 -6.267e- 12

2 2.715e+1 -5.404e-I 2.950e-3 2.723e-7 -9.338e-9 9.968e-12

3 -3.486e+1 1.113e+0 -1.301e-2 6.496e-5 -1.182e-7 8.042e-11

4 -6.051 e+ I 1.409e+0 - 1.208e-2 4.405e-5 -5.674e-8 2.566e- 11

5 -2.669e+1 5.283e-1 -3.445e-3 6.072e-6 1.252e-8 -2.155e-11

6 -6.727e+0 4.226e-2 1.044e-3 - 1.292e-5 4.740e-8 -4.486e- 11

7 1.879e+1 -5.836e- 1 6.968e-3 -3.939e-5 1.012e-7 -8.230e- 11

8 1.034e+2 -2.513e+O 2.375e-2 - 1.069e-4 2.184e-7 - 1.370e- 10

9 - 1.048e+1 3.821e-1 -5.227e-3 3.441e-5 -1.108e-7 1.409e- 10

10 1.677e+0 6.540e-2 -1.813e-3 1.291e-5 -2.672e-8 1.979e-11

3.4 Sensitivity of Upward Flux to Surface Temperature

The parameterization also computes the sensitivity of upward fluxes at all levels to the

surface temperature 0s. In a general circulation model, the LW parameterization is called

relatively infrequently (every few hours), whereas the boundary layer and land surface

parameterizations use time steps of a few minutes. To maintain consistency between the upward

radiative fluxes aloft and that at the surface when the surface temperature changes significantly

between calls to the LW parameterization, all fluxes are linearized about the surface temperature

at the beginning of the radiation interval, and radiative heating rates are recomputed based on this

linearization every few minutes, at each time step. The partial derivative of the upward flux with

respect to the surface temperature is computed from

= li(P'Ps)l _ I = Ti(p,ps)ei Ci,l+ nci nOsn-1

_0 s L ov _10_-Os n=2 '
4. BAND-AVERAGED TRANSMISSION FUNCTIONS OF GASES

For accuracy and speed considerations, the flux transmittance defined by (3.7) is computed

using three different approaches, depending on the absorber and the spectral band:

The k-distribution method with linear pressure scaling is applied to the absorption due to

water vapor and minor trace gases. This method is also applied to the C02 absorption in

Band 3 if accurate cooling rate calculations in the middle atmosphere (p < 10 hPa) are not

required.

The transmittances due to CO: and 03 absorption in Bands 3 and 5 are obtained from precomputed transmittance tables. Because the k-distribution method with linear pressure scaling

underestimates the water vapor cooling rate in the middle atmosphere, the transmittances of

the three strongest water vapor absorption bands (Bands 1, 2 and 8) are also derived from precomputed transmittance tables if accurate computations of the water vapor cooling in the

middle atmosphere are required.

• The transmittances due to water vapor continuum absorption in Bands 3-7 are computed

using a one-parameter scaling approach.

Applications of these parameterizations to the different spectral bands and absorbers are

summarized in Table 1.

4.1 The k-distribution method

As shown in Chou et al. (1993), the cooling due to water vapor in the lower atmosphere (p

> 10 hPa) is primarily attributable to the spectral regions away from the center of absorption

lines. The absorption coefficient is approximately linear in pressure, and its dependence on

temperature varies smoothly with wavenumber. Under these conditions, the absorption

coefficient at any temperature and pressure is simply proportional to its value at a reference

pressure and temperature. The absorption coefficient can, therefore, be extrapolated from the

absorption coefficient at a reference pressure and temperature according to

kv(p, O) = kv(pr, Or) (P / Pr) " h (0, Or) (4.1)

where p, and 0r are the reference pressure and temperature, m is an empirical constant, and h(O,

Or) is the temperature scaling factor, which we fit by

8
h (O, O_)=l+a(O-O,)+_(e- Or)2 (4.2)

The regression coefficients _ and 13are derived for each spectral band to minimize errors m flux

transmittances. Details of the derivation of the coefficients c_ and [3 are given in Chou et al.

(1993). Values ofp, 0r, m, _, and [3 are given in Table 3 for scaling the water vapor and CO:

absorption coefficient.

Table 3. Spectral bands, reference pressure and temperature, and the coefficients for the

pressure and temperature scaling used in the k-distribution method. Band 3 is divided into three

sub-bands.

Band

He0 pr = 500 hPa 0r= 250 K CO, 0r= 250 K

m or(K-') 9(K-z) Pr (hPa) m a(K-') 9(K-2)

1 1.0 .0021 -1.01e-5

2 1.0 .0140 5.57e-5

3a 1.0 .0167 8.54e-5 300 0.50 .0179 1.02e-4

3b 1.0 .0167 8.54e-5 30 0.85 .0042 2.00e-5

3c 1.0 .0167 8.54e-5 300 0.50 .0184 1.12e-4

4 1.0 .0302 2.96e-4

5 1.0 .0307 2.86e-4

6 0.77 .0195 1.1 le-4

7 0.5 .0152 7.61e-5

8 1.0 .0008 -3.52e-6

9 1.0 .0096 1.64e-5

10 1.0 .0149 6.20e-5 300 0.50 .0179 1.02e-4
Withthescalingof(4.1),themonochromatic fluxtransmittance(2.3)canbeexpressedas

Tv(p,p')= 2 e-k_(Prt_rlw(p'P')4agdg (4.3)

where w is the scaled absorber amount given by

w( p, p') = c(p" ) _' ' lbr)rn h (0" Or )dp" (4.4)

t

and c is the absorber mass concentration. It can be seen from (4.3) and (4.4) that the dependence

of Tv on wavenumber is separated from its dependence on pressure and temperature.

Wavenumbers which have the same absorption coefficient at Pr and 0r will have the same

transmittance at all other pressures and temperatures. Within a narrow spectral interval where the

Planck function can be considered constant, these wavenumbers are radiatively identical, and the

integration over wavenumber can be replaced by an integration over the absorption coefficient k,

flay ( J0 ( )f(k)dk )dv / Av

wheref(k) is the k-distribution function at the reference pressure Pr arut temperature Or, and

(4.5)

k) dk = 1 (4.6)

This is the basic assumption of the k-distribution method. It should be noticed that the

approximation for relating the absorption coefficient at a pressure level to that at other levels, as

shown in (4.1), is different from the correlated-k distribution method (e.g. Wang and Shi, 1988;

Goody et al., 1989; Lacis and Oinas, 1991; Fu and Liou, 1992; Mlawer et al., 1997). However, it

is also a type of the correlated-k approximation with a well-defined correlation.

If we divide a wide spectral band into a number of narrow sub-bands, for which the Planck

function can be considered constant, and apply the k-distribution method to each sub-band, the

Planck-weighted transmission function given by (3.7) becomes

10
Z[ j 0o f 1

j L ,/By)

T(w) =

_0o )

Z[Bi(Oo)'Svjfk Tk(W)fj(k)dk]

J

B(Oo)

(4.7)

where

Tk (w) = 2 e-kw/la fad_t (4.8)

Bj(Oo) is the mean Planck function of the sub-band j, and fj(k) is the k-distribution function of the

sub-band at the reference pressure p, and temperature 0,.

The flux transmittance given by (4.8) can be accurately computed by using the following

approximation (see Figure 5b of Chou and Arking, 1980)

Tk(w) _ e-kW/_ (4.9)

where 1 /_" is the diffusivity factor taken to be 1.66. Using this approximation and replacing the

k integral by a simple quadrate, Equation (4.7) reduces to

N

T(w) = Ze kn v*/_ A g n (4.10)

I

n=l

where Ag. is the Planck-weighted k-distribution function for the n'hk-interval given by

Bj(0o) 8v

Agn= Lfj(kn)_kn B(Oo) j (4.11)

J

Values of Ag, are first derived from line-by-line calculations and then slightly adjusted using

regression so that the rms difference between the transmittances computed from the line-by-line

method and from (4.10) is minimized. (The line-by-line calculations are briefly discussed in

Section Section 9). The adjustment is necessary because of the small number of terms (N = 6)

used and the diffusivity (4.9) approximation applied. Details are given in Chou et al. (1993).

11
Calculations of the Planck-weighted flux transmission function using (4.10) are very fast

for two reasons: (1) For each k, it requires only L exponential operations for calculating

e -k_'_/_, l = 1, 2, .... , L, where L is the total number of atmospheric layers, and Awt is the scaled

absorber amount in the layer 1. The transmittance of a path consisting of more than one layer can

be obtained by multiplying the transmittances of individual layers. By comparison, the

commonly used band models require at least 0.5 L 2 exponential operations. (2) We choose values

of k in such a way that

k. = 1] k._l n=2, 3..... N (4.12)

where 11 is a positive integer. With this choice of k's, only a single set of L exponential operations

for the first value of k is needed. The other exponential terms can be derived from that of the

previous value of k, with each derivation involving only -3 multiplication operations. Because a

multiplication operation is much faster than an exponential operation, flux calculations can be

greatly accelerated by choosing values of k's according to (4.12) for computing transmittances.

The first absorption coefficient, k_, the constant rl, and the k -distribution function are given in

Table 4 for the water vapor.

Table 4. Parameters for the transmittance given by (4.10) due to water vapor line absorption.

k I / _ is the first absorption coefficient, and 11 is the constant given in (4.12). Ag is the kdistribution function (4.1 i ). Band 3 is treated separately in Section 4.5. Units of k are g-Zcm 2.

Band 1 Band2 Band4 Band5 Band6 Band7 Band 8 Band9 Band 10

kl/_ 2.90e+1 4.17e-! 5.25e-4 5.25e-4 9.37e-3 4.72e-2 1.32e+0 5.25e-4 1.06e-I

6 6 6 6 8 9 6 16 8

Ag_ .2747 .1521 .4654 .5543 .5955 .1958 .0740 .1437 .3153

Ag 2 .2717 .3974 .2991 .2723 .2693 .3469 .1636 .2197 .4604

Ag_ .2752 .1778 .1343 .1131 .0953 .3147 .4174 .3185 .1326

Ag4 .1177 .1826 .0646 .0443 .0335 .1013 .1783 .2351 .0798

Ag5 .0352 .0374 .0226 .0160 .0064 .0365 .1101 .0647 .0119

Ag6 .0255 .0527 .0140 .0000 .0000 .0048 .0566 .0183 .00013

12
MINOR ABSORPTION BAND

In addition to the major absorption due to water vapor, CO2 and 03, there are also many

weak absorption bands of N20, CH4, C02, and CFCs located mostly in the water vapor window

region between 800 and 1300 cm _. Individually, the effect of these bands on fluxes is small, but

collectively the effect is to reduce the outgoing LW radiation at the top of a clear atmosphere by

-4 W m 2 and to enhance the surface heating by -2 W m -2. These effects are significant in the

study of anthropogenic climate change.

The spectral bands which include the minor absorption are shown in Table 1. A total of 13

sets of minor transmission functions are computed: four in Band 4, two in Band 5, four in Band 6,

two in Band 7, and one in Band 10 (Sub-band 3a).

The k-distribution method is also applied to computing the transmission functions averaged

over individual minor spectral bands (Table 1). The reference pressure and temperature, values of

the parameters used for the minor bands are listed in Tables 5-7 (see Kratz et al., 1993, and Kratz

et al., 1998, for details of the parameterization). Because the absorption is weak in those minor

bands, it requires only a small number (1-4) of the k-intervals for accurate transmission

calculations.

13
Table 5. Parameters for computing the flux transmission functions due to N20 and CH 4

using the k-distribution method. (pr= 500 hPa and 0,= 250 K).

N20 CH 4

Band 6 Band 7 Band 10 Band 6 Band 7

/,I / ta 6.32e-2 5.36e-2 2.52e-1 5.81e-3 6.29e-2

q 21 8 58 12

Ag_ .9404 .5620 .971 i .000 .6107

Ag: .0596 .1387 .029 .2802

Ag. .2406 .1073

AX', .0587 .0018

m .0 .48 .0 .0 .65

1.93e-3 1.38e-3 1.45e-3 1.70e-2 5.96e-4

13 4.38e-6 7.48e-6 3.67e-6 1.58e-4 -2.29e-6

Table 6. Same as Table 5, except for the absorption due to C02.

Band 4 Band 5 Band 10"

kl / lu 1.92e-7 1.92e-7 2.66e-5

1"1 5 5 8

Ag / .1216 .0687 .267

Ag2 .2436 .1480 .220

Ag3 .2498 .1951 .210

Ag 4 .2622 .3344 .241

Ag 5 .0781 .1720 .020

Ag6 .0427 .0818 .042

m .0 .0 .5

ot 3.58e-2 3.43e-2 1.79e-2

4.05e-4 3.74e-4 1.02e-4

* Pr = 300 hPa.

14
Table7. SameasTable5,exceptfortheminorCFCabsorptionbands.Thereisonlyone

k-interval for each CFC band, and there is no pressure scaling applied, i.e. m=O.

CFC-11 CFC-12 CFC-22

Band 4 Band 5 Band 4 Band 6 Band 4 Band 6

k-l- 1.90e+l 1.02e+l 1.58e+1 3.70e+1 6.18e+0 3.28e+1

la

c_ 1.27e-3 8.19e-4 8.77e-4 8.62e-4 9.65e-4 -3.00e-5

13 3.56e-6 4.68e-6 -5.88e-6 -4.23e-6 1.31e-5 5.25e-7

4.2 Pre.computed Transmission Tables

Although the k-distribution method with linear pressure scaling is computationally very

fast, it is not accurate in the middle atmosphere (0.01-10 hPa), where the pressure ranges by 3

orders of magnitude and where the Doppler broadening of absorption lines is important. Cooling

in the middle atmosphere is primarily due to CO 2 in the 15-pro band (Band 3), 03 in the 9.6-pm

band (Band 5) and secondarily due to water vapor near the centers of absorption bands (Bands 1,

2, and 8).

It is shown in Chou and Kouvaris (1991) and Chou and Suarez (1994) that transmittances

in these bands can be simply derived from pre-computed transmission tables. The basis for this

method is that a nonhomogenous layer with pressure and temperature varying with height can be

treated as if it were homogeneous with an effective pressure and temperature defined by

S Pdu

Perf- S d u (4.13)

S Odu

(4.14)

Oeff-- Sd u

15
where u is the absorber amount, and the integration is over the depth of a layer. It has been well

recognized (e.g. Wu, 1980; Chou and Kouvaris, 1991) that the contribution to the atmospheric

cooling is primarily from near-by layers. The pressure and temperature variations among near-by

layers are small, and the simple scaling approximations of (4.13) and (4.14) can be applied to

accurate calculations of cooling rate.

With the two-parameter scaling of (4.13) and (4.14), the flux transmission becomes a

function of the absorber amount and the effective temperature and pressure. These dependencies

can be accurately pre-computed from the following equation using a line-by-line method,

T( u,peff ,Oeff )= _A¥

Tv (U,Peg,Oeff)B,,(0o)dV

fAv_ (0o)av

(4.15)

where Av is the width of the entire spectral band. The band transmission varies rapidly with

pressure, but rather smoothly with temperature. The size of the three-dimensional transmission

tables can be reduced to three two-dimensional tables using a quadratic fit in temperature,

T( U,peff ,Oeff ) = a( u,p eff ) + b( u,peff )(Oeff - 250) + c( u, Peff )(Oeff - 250) 2 (4. ! 6)

where a, b, and c are regression coefficients. This regression is valid for temperatures ranging

from 170K to 330K; for this range it introduces only - 1% error in the absorptance. Table 8

shows the first values, the intervals, and the interval numbers of logm(u) and logto(PeH) of the

tables a, b and c for transmissions due to CO 2 absorption in the 15/on region, O_ absorption

in the 9.6 _ region, and water vapor absorption in the three strong absorption bands. It is

noticed that the transmission tables can be applied to an atmosphere with a CO2 concentration 100

times of the present value, i.e. 35000 ppmv.

The absorption due to 03 covers a spectral range wider than that of Band 5 shown in Table

1. To include the weak absorption outside this spectral band, we have made adjustment to the

transmission function by requiring that

(l-x) Be (T)dv = _! -Xv )Be (T)dv

1

(4.17)

16
Table8. Thefirst values,loglo(w,_)l and loglo(p,_)_, the intervals, Alogto(Welr) and

Alog,o(p,lr), and the numbers, nw and np, of logm(w,lr) and logto(P_st) of the tables a, b

-2

and c for transmittance calculations using (4.16). Units of w,_ are g cm for water

vapor and (cm-atm)sre for COz and 03, and the unit ofp_ is hPa.

Band 1 Band 2 Band 3 Band 5 Band 8

Absorber H20 H20 CO 2 03 H20

logl_[w_)l -8 -8 -4 -6 -8

lOglo(Pelr)1 -2 -2 -2 -2 -2

Alog_o(We_) 0.3 0.3 0.3 0.3 0.3

Aloglo(p4y) 0.2 0.2 0.2 0.2 0.2

nw 31 31 30 21 31

np 26 26 26 26 26

where v_ - v2 is the spectral range of Band 5, va - v_ is a wider spectral range that covers all the

molecular lines in the absorption band, and "c is the effective mean transmission of Band 5. Thus,

x includes the effect of the O3 absorption outside the spectral range v, - vb. We could have

expanded the range v_ - v2, but the accuracy of the transmission calculations would be reduced

when (4.27) is applied to include the water vapor effect, which is discussed in Section 4.4. The

adjusted mean transmission is then pre-calculated from

_ __vb (1-x v )/_,

(T)dv

x = 1- _ (4.18)

'vlZBy (T)dv

The method of table look-up is simple and accurate. As shown in Chou and Kouvaris

(1991) and Chou and Suarez (1994), it can be applied to accurate calculating the fluxes and

cooling rate in both the middle and lower atmosphere extending from 0.01 hPa to the earth's

surface. However, this method could be significantly slower than the k-distribution method on

computers that cannot efficiently perform table look-ups. We have therefore provided the option

17
ofusingthek-distribution method for all bands, except for the computation of O_ transmission in

Band 5.

4.3 One-parameter scaling for water vapor continuum absorption

As shown in Table I, water vapor continuum absorption is included in Bands 3-7. The

water vapor continuum absorption coefficient increases with increasing water vapor partial

pressure but with decreasing temperature. It can be approximated by

18oo(-1-±)

kv(pe,O)=kv,olPele k0 296] (4.19)

[.)Po

where Pe is the water vapor partial pressure in hPa, po = 1013 hPa, and kv,o is the absorption

coefficient when p_ = 1013 hPa and 0 = 296K. Values of kv,o are computed from the analytical

representation given by Roberts et al. (1976), which is a fit to the laboratory data of D. E. Burch

for Bands 3-6 and from Clough et al. (1989) for Band 7.

With the scaling of (4.19) for the absorption coefficient, the flux transmission reduces to

Tv(p,p') = 2 e-kv'°wC(p'P')Ogd_ (4.20)

where w" is the scaled water vapor amount for continuum absorption given by

wC(p'P') =Tfp' q2(p,,_ O(p") 296 dp" (4.21)

Here we have used the approximation

qP

Pe - (4.22)

0.622

and q is the specific humidity.

Taking the Planck-weighted average of (4.20) over a band we have

T(w c) = V (4.23)

_a,, _' (0o)_

18
wherewenotethatthetransmissionis a function of wc only. The absorption coefficient kv.o varies

slowly with wavenumber, except in Band 3 where it varies by a factor of =3.5. Therefore, we

divide Band 3 into three sub-bands (see Section 4.5 for more detail of the division of Band 3 into

3 subbands). The Pianck-weighted flux transmissions for the three sub-bands and for Bands 4 - 7

were computed from (4.23). These transmittances were then fit by

T( w c) = e -U wc/_ (4.24)

The effective absorption coefficients, k c / _t, for water vapor continuum absorption are given in

Table 9.

Table 9. The effective absorption coefficient k c/'_

absorption for Bands 3-7. Units of k c / "_ are g l cm 2.

for the water vapor continuum

Band3a Band3b Band 3c Band4 Band5 Band6 Band7

k c

--=- 109.6 54.8 27.4 15.8 9.40 7.75 8.78

_t

4.4 Overlapping of Gaseous Absorptions

When there are more than one absorber involved in a spectral band, overlaps must be

considered. The total transmission for, say, two absorbers, can be written as

T,2-JT l(V)T 2(v)dv

f dv (4.25)

where the subscripts 1 and 2 denote the two absorbers. If the transmittance is expressed as the

sum of the mean over the spectral band, T, and the deviation from the mean, T', then (4.25)

reduces to

19
dv =TI T2+ _dv

f T' 1 (v)T'2 (v)dv

(4.26)

If the overall shapes of the absorption curves due to both absorbers are uncorrelated, the

last term of (4.26) can be neglected, and the total transmittance becomes

-- I

Tl2 = Ti T2 (4.27)

4.5 Special Treatment of the 15-1tm Band

The 15-/_m band poses additional difficulties for two reasons. First, the water vapor line

absorption and the continuum absorption are highly correlated. As can be seen in the bottom two

panels of Figure 2, the absorption coefficient increases with decreasing wavenumber by a factor

of about 100 for water vapor line absorption and about 10 for continuum absorption. Thus, the

water vapor line and continuum absorption are highly correlated, and the approximation (4.27)

cannot be applied directly to the entire band. Second, the C02 absorption coefficients differ by

several orders of magnitude between the band center and the wings (see the top panel of Figure

2). Rather than trying to parameterize the correlation effect or the variations in CO 2 absorption,

we simply divide the band into three sub-bands (see the bottom panel of Figure 2 and Table 1)

and then combine the parameterized transmittances of the sub-bands into a single band

transmittance. This transmittance is then used in the usual way to solve the transfer equations for

the entire band.

20
I

i

CQ

2

'_'_L,_, _, Lh,E

'i',_L_, V'_la, CanLlquum

5dC, 5F=O SBO _ £20 F=,40 £I_0 f>BO 7(_0 TZO TdO TI_D TBO

WavenuI_ber (cn'i_)

Figure 2. Schematic of the absorption coefficient for C02 and for water vapor line and

continuum absorption in the 15-pro region (Band 3). The subdivision of the band into three subbands is made to accurately account for the large variations of the absorption coefficients.

The H20 line absorption in Band 3 is computed from the k-distribution method. We divide

the band into three sub-bands. Within each of the three sub-bands, the transmittances due to line

and continuum water vapor absorption are sufficiently uncorrelated that we can overlap them

using the multiplication approximation (4.27). Letting TaL and Tic be the line and continuum

transmittances of sub-band i, we obtain the total water vapor transmittance in Band 3 by

overlapping them and taking the Planck-weighted average

3 B.(O )

Twv= Z T.Lz.CI- "--Lz.z_] (4.28) ,' ' LS(Oo)J

i=l

where B/Oo) is the Planck flux integrated over Sub-band i, and B(Oo) is the Planck flux integrated

over the three sub-bands.

21
Substituting for the line transmittance from (4.10) and for the continuum transmittance

from (4.24), we have

3 c - N -k , [ B,mo)q

TWV=Ee-ki"w IlaEe ['ag)n,i[_ j

i= 1 n=l

3 _N

i= 1 n=l

(4.29)

We use a single set of kt, and 1"1for the three water vapor sub-bands. See (4.12) for relating kl to

k,. Table 10 shows the values of these parameters. The pressure and temperature scaling

parameters are shown in Table 3. Since we use the same scaling parameters for the three subbands, the scaled water vapor amounts, w and w c, are independent of sub-band. Therefore, the

exponentials for the water vapor line transmittance in the inner summation of (4.29) only need to

be evaluated once.

Table 10. The first value of the absorption coefficient k I / _ for the water vapor line

absorption, the value of rl that k, = rlk,__, and the Planck-weighted k-distribution

function (A_) given by (4.29) and (4.30) of the three sub-bands in the 15 pm region.

WATER VAPOR CO2

Subband a Subband b Subband c Wings Center

k 1/ _ 1.33e-2 1.33e-2 1.33e-2 2.66e-5 2.66e-3

lq 8 8 8 8 8

(A_)I .1782 .0923 .0000 .1395 .0766

(A_)2 .0593 .1675 .1083 .1407 .1372

(A_) 3 .0215 .0923 .1581 .1549 .1189

(Ag) 4 .0068 .0187 .0455 .1357 .0335

(A_) 5 .0022 .0178 .0274 .0182 .0169

(Ag) 6 .0000 .0000 .0041 .0220 .0059

The continuum absorption coefficients, kc, are given in Table 9. Note that the coefficients

for Sub-bands 3a and 3b are integer multiples of the coefficient for Sub-band 3c; thus only one

22
exponentiation for the water vapor continuum transmittance needs to be performed in evaluating

T wv for the three sub-bands.

The C02 transmittance may be computed either using the k-distribution method or using

table look-up. When the table look-up is used, a single transmittance is computed directly for the

entire band. When the k-distribution method is used, we separate Sub-band b (the center region)

from Sub-bands 3a and 3c (the wings). The optical properties of Sub-bands 3a and 3c are very

similar, and so we have combined them by using the same scaling parameters (p,, 0,,o., _, and m

in Table 3 and k_ in Table 10). The band-averaged CO2 transmittance for Band 3 is thus computed

from

N N

TC02 = Ze-(kc),u_/_/- u-Xgc)n ^ _ + Ze--(kw), u_.lit (A_w)n (4.30)

n=l n=l

where the subscripts c and w denote, respectively, the band center (Sub-band 3b) and band wings

(Sub-band 3a and 3c), and u,. and u_ are the scaled C02 amount in the band center and band

wings. Note that

N

(A_c)n +(A_w)n = 1 (4.31)

n=l

Values of (A_) n are given in Table 10.

Finally, the total flux transmittance in the 15-1am band is computed from

T = TWVT c°2 (4.32)

5. SPECIAL TREATMENT OF THE 17-,um NzO BAND

The spectral range of the absorption due to N20 in the 17-/nn region is relatively narrow

(560-620 cm 1) and coincides approximately with the Sub-band 3a. As shown in Figure 2, the

absorption in this subband differs significantly from Sub-bands 3b and 3c. Therefore, the

absorption due to N20 in this band is treated differently from the others and is designated as Band

10 shown in Table 1. Only the changes in fluxes due to the N20 absorption are computed. From

(3.9) and (3.10), the change in fluxes can be computed from

23
AF3a $ (P) = f0 '_B3a (O') 0[ AT3_bp'(p,p') ] dp' (5.1)

f p' O[AT3a(P'P') ]

AF3a $ (p) =eiB3a(Os)AT3a(p,ps) + B3a(O') dp'

_ 3P'

(5.2)

In the above equations, the change in the mean transmittance due to N20 is defined as

AT3a=T * 3a-_a=l _L 3al_C 3al_CO_"3a[1[_N_O_ 3a-! ]

(5.3)

where the subscript 3a denotes Sub-band 3a, the superscript * denotes inclusion of the N20

absorption, and the superscripts L, C, CO2 and N,O denote water vapor line, water vapor

continuum absorption, CO, and N20 respectively. The upward and downward fluxes are then

adjusted according to

.$

F (p)= F$(p)+ AF3a$(p) (5.4)

F*'I'(p) = F$(p)+ AF3a $ (p) (5.5)

The transmission functions are computed using the k-distribution method (4.10) for T L,

T C02 and T N20, and the one-parameter scaling method (4.24) for T C. The parameters used for

computing the transmission functions of Band 10 are given in Tables 3-6.

6. CLOUDS

6.1 Cloud Single-Scattering Properties

Scattering by clouds is parameterized separately for water droplets and ice particles. The

high-spectral resolution extinction coefficient, the single-scattering albedo, and the asymmetry

factor were computed from the Mie theory for water droplets and the method of Fu and Sun

(1998) for hexagonal ice particles. Dr. Si-Chee Tsay of NASA/Goddard Space Flight Center

provided us data of these parameters for water drops, and Prof. Qiang Fu of the University of

Washington provided us the data for ice particles. Mean values of the extinction coefficient 13,

single-scattering albedo co, and asymmetry factor g are in turn derived for each spectral band

according to

24
Av Av

(6.1)

Av Av

(6.2)

gl r,.) = Zgv(re)_v(re)_Jv(re)Bv(eo)lZO_v(re)_(re)t_(Oo)

Av Av

(6.3)

whcrc r, is thc cffcctive size of cloud particles which is a function of the ratio of the particle

volumc to thc surface area, B is the Pianck function, 0o is a medium value of the atmospheric

temi_'raturc _t to bc 250 K, and Av is the spectral interval of a band. We fit the mean singlescattering paramctcrs by the following functions of r,,

Cl3,2 for ice particles

[_(reJ = c[3.1+ (re)CB.3

(6.4a)

= Z cf_,i re i-I

i=l

for liquid drops (6.4b)

4

re(re) = Z¢co,irei-I

i=1

for both ice and liquid water particles (6.5)

4

g(re) = ZCg,i rei- 1

i=1

for both ice and liquid water particles (6.6)

Tables 11-13 list values of the regression coefficients, c_, c,_ cu, for both water droplets and ice

particles. Finally, the cloud optical thickness is derived from

x= 13 CW-P (6.7)

where CWP is the cloud water path (column mass per unit area), either ice or liquid water. In the

current version of the code, the effective radii re are input parameters and are not computed

within the code.

25
Table11a.Coefficientsforcomputingtheextinctioncoefficientforcloud

iceparticlesfrom(6.4a).Unitsof theparticlesizein(6.4a)are_tm.

Band c1_.1 c 1_2 c 1_3

1 -0.4417 ! 0.62951 0.06465

2 -0.13727 0.61291 0.28962

3 -0.01878 1.67680 0.79080

4 -0.01896 1.06510 0.69493

5 -0.04788 0.88178 0.54492

6 -0.02265 1.57390 0.76161

7 -0.01038 2.15640 0.89045

8 -0.00450 2.51370 0.95989

9 -0.00044 3.15050 1.03750

10 -0.02956 1.44680 0.71283

Table 1lb. Coefficients for computing the extinction coefficient for cloud

liquid water droplets from (6.4b). Units of the particle size in (6.4b) are _m.

Band cl__ c _2 c _3 c _4

1 0.08641 0.01769 -1.5572E-3 3.4896E-5

2 0.22027 0.00997 -1.8719E-3 5.3112E-5

3 0.38074 -0.03027 1.0154E-3 - 1.1849E-5

4 0.15587 0.00371 -7.7705E-4 2.0547E-5

5 0.05518 0.04544 -4.2067E-3 1.0184E-4

6 0.12724 0.04751 -5.2037E-3 1.3711E-4

7 0.30390 0.01656 -3.5271E-3 1.0828E-4

8 0.63617 -0.06287 2.2350E-3 -2.3177E-5

9 1.15470 -0.19282 1.2084E-2 -2.5612E-4

10 0.34021 -0.02805 1.0654E-3 - 1.5443E-5

26
Table12a.Coefficientsfor computingthesingle-scatteringalbedofor

cloudiceparticlesfrom(6.5).Unitsoftheparticlesizein(6.5)arepro.

Band ca_ c o_2 c _._ c _4

l 0.17201 1.2229E-2 -1.4837E-4 5.8020E-7

2 0.81470 -2.7293E-3 9.7816E-8 5.7650E-8

3 0.54859 -4.8273E-4 5.4353E-6 - 1.5679E-8

4 0.39218 4.1717E-3 -4.8869E-5 1.9144E-7

5 0.71773 -3.3640E-3 1.9713E-5 -3.3189E-8

6 0.77345 -5.5228E-3 4.8379E-5 - 1.5 ! 51E-7

7 0.74975 -5.6604E-3 5.6475E-5 -1.9664E-7

8 0.69011 -4.5348E-3 4.9322E-5 -1.8255E-7

9 0.83963 -6.7253E-3 6.1900E-5 -2.0862E-7

I 0 0.64860 -2.8692E-3 2.7656E-5 -8.9680E-8

Table 12b. Coefficients for computing the single-scattering albedo for

cloud liquid water droplets from (6.5). Units of the particle size in (6.5)

are lam.

Band c_1 c 0_2 c,_s c o_4

1 0.17201 1.2229E-2 - 1.4837E-4 5.8020E-7

2 -0.01338 9.3134E-2 -6.0491E-3 1.3059E-4

3 0.03710 7.3211E-2 -4.4211E-3 9.2448E-5

4 -0.00376 9.3344E-2 -5.6561E-3 1.1387E-4

5 0.40212 7.8083E-2 -5.9583E-3 1.2883E-4

6 0.57928 5.9094E-2 -5.4425E-3 1.2725E-4

7 0.68974 4.2334E-2 -4.9469E-3 1.2863E-4

8 0.80122 9.4578E-3 -2.8508E-3 9.0078E-5

9 1.02340 -2.6204E-2 4.2552E-4 3.2160E-6

10 0.05092 7.5409E-2 -4.7305E-3 1.0121E-4

27
Table13a.Coefficientsforcomputingtheasymmetryfactorforcloudice

particlesfrom(6.6).Unitsof theparticlesizein(6.6)are_tm.

Band Cg,l c_e Cg,3 Cg,4

I 0.57867 1.0135E-2 -1.1142E-4 4.1537E-7

2 0.72259 3.1149E-3 -I .9927E-5 5.6024E-8

3 0.76109 4.5449E-3 -4.6199E-5 1.6446E-7

4 0.86934 2.7474E-3 -3.1301E-5 1.1959E-7

5 0.89103 1.8513E-3 -1.6551E-5 5.5193E-8

6 0.86325 2.1408E-3 -1.6846E-5 4.9473E-8

7 0.85064 2.5028E-3 -2.0812E-5 6.3427E-8

8 0.86945 2.4615E-3 -2.3882E-5 8.2431E-8

9 0.80122 3.1906E-3 -2.4856E-5 7.2411E-8

10 0.73290 4.8034E-3 -4.4425E-5 1.4839E-7

Table 13b. Coefficients for computing the asymmetry factor for cloud liquid

water droplets from (6.6). Units of the particle size in (6.6) are lam.

Band cg, j Cg,2 Cg,3 Cg,4

! -0.51930 0.20290 -1.1747E-2 2.3868E-4

2 -0.22151 0.19708 - 1.2462E-2 2.6646E-4

3 0.14157 0.14705 -9.5802E-3 2.0819E-4

4 0.41590 0.10482 -6.9118E-3 1.5115E-4

5 0.55338 7.7016E-2 -5.2218E-3 1.1587E-4

6 0.61384 6.4402E-2 -4.6241E-3 1.0746E-4

7 0.67891 4.8698E-2 -3.7021E-3 9.1966E-5

8 0.78169 2.0803E-2 - 1.4749E-3 3.9362E-5

9 0.93218 -3.3425E-2 2.9632E-3 -6.9362E-5

10 0.01649 0.16561 -1.0723E-2 2.3220E-4

28
6.2 Layer optical properties

When the absorption and scattering due to clouds interact with that of gases and aerosols in

a layer, the effective optical thickness, single-scattering albedo, and asymmetry factor in a

spectral band are computed from

x = '_,'_i (6.8)

i

CO-ZZ= (OiZ i ] ,1:i (6.9)

i i

g=Zgi(OiTi/Zf.Digi (6.10)

i i

where the summation is over all gases, aerosols, and liquid water and ice particles, i. Thus, in the

LW radiation code, ice and liquid water cloud particles are allowed to mix in a layer.

6.3 Paramaterization for Cloud Scattering

Explicit calculations of scattering by clouds requires a large amount of computer time.

Instead, we scale the cloud optical thickness 't to approximate the effect of scattering according

to,

x c =(1-oafyr (6.11)

where f is the fraction of radiation scattered downward (upward) for radiation incident from

above (below). This scaling takes into account the effect of back-scattering and absorption by

cloud particles.

Assuming that the scattering phase function can be approximated by the HenyenGreenstein function, the forward-scattering function fis computed as a function of the asymmetry

factor and fit by a polynomial function

4

f = Zaig i-I (6.12)

i=1

where al = 0.5, a2 = 0.3738, a3 = 0.0076, and a4 = 0.1186.

29
Thus, the cloud optical thickness is reduced from x to "tc, and scattering is not explicitly

computed. By scaling the cloud optical thickness (6.1 l), the effect of scattering is parameterized

but not explicitly computed. The computing time is the same as the case without taking into

account the cloud scattering. See Chou et al. (1999) for details of the parameterization for the

cloud LW scattering.

6.4 Cloud Overlapping

Equation (2.3) applies to the case where the atmosphere is horizontally homogeneous.

When clouds with fractional areal cover occur at different heights, assumptions have to be made

on the ways these clouds are overlapped for radiative flux calculations. There are a number of

schemes used to vertically associate clouds at different height (e. g. Manabe and Strickler, 1964;

Geleyn and Hollingsworth, 1979; Liang and Wang, 1997; Raisanen 1998; Li., 2000). These

schemes all assume that clouds either are randomly overlapped or have a combination of random

and maximum overlapping. In our radiation parameterization, we also adopt a maximum-random

overlapping cloud scheme.

If a cloud layer with fractional cover A and optical thickness "of is introduced between p

and p', the mean radiance transmittance in the direction la becomes

[I- A ]e -'_v 'p' p' ) _ + Ae -['tv (p' p') +xv c( p'p' ]/lt

and the mean flux transmittance is

1

* s{ t

Tv (p,p):2 [1-A]e-X"(P'P')O+Ae -[%(p'p')+%_]/I't I.td_l

0

=[l-A(l-e -%c/_)]T v (P,P')

(6.13)

m

where x_ is the optical thickness of a clear atmosphere, g is the effective mean value of/l which

converts the radiance transmittance to flux transmittance, and Tv is the transmittance of a clear

atmosphere defined by (2.3). The diffusivity factor 1 /_" depends on % but is commonly chosen

m

to be 1.66. Letting Nv = A (1- e -% c/la ), Equation (6.13) becomes

Tv *(p,p') = (1- Nv )Tv (p,p') (6.14a)

30
If wefurtherassumethatXv' is constant within a spectral band and integrate Equation (6.14a) over

the band, we have

T*( p,p') = (1- N)T( p,p') (6.14b)

where T is the band-averaged flux transmittance defined by (3.7). Because 1 - e -xc/_ is the flux

emissivity of the cloud layer, N can be regarded as the effective cloud cover with an emissivity of

1. Alternatively, (l-N) can be regarded as the effective cloud transmissivity with a fractional

cloud cover of 1.

When there is more than one cloud layer with fractional cover between p and p', the

situation is considerably more complicated since we need to describe how the clouds are

overlapped. In general we can write

T*( p, p') = C(p, p') T (p, p') (6.15)

where C(p,p') depends on the values of A and x c for the various cloud layers.

For the special case of any combination of overcast and randomly overlapped fractional

cloud layers, we have

C(p,p') = H (1- Nj )

(6. 16)

J

where the subscript j denotes the cloud layers between p and p'. Since 17[(1 - Nj) is the fraction

of the horizontal area that would be cloud-free if all cloud layers (including overcast layers) were

assigned their equivalent black-cloud fraction and then randomly overlapped, Harshvardhan et al.

(1987) referred to C(p,p') as the probability of a clear line-of-sight between p and p'. Obtaining

C(p,p') for non-random arrangements of gray clouds, however, is not as straightforward. In

general, C(p,p') does not depends on N alone, but on more complicated combinations of A and

'17c"

Consider another arrangement discussed by Harshvardhan et al. (1987), that of maximally

overlapped clouds. If the largest cloud between p and p' is black, then C(p,p') is simply equal to

one minus its fraction. If all clouds are allowed to be gray, however, C(p,p') will depend on all

cloud fractions and optical thicknesses. For the maximally overlapped clouds between p and p',

31
the clear line-of-sight C(p,p') is independent of the order of cloud layers and may be computed by

putting the clouds in order of increasing cloud cover:

For the special case with three maximally overlapped clouds between p and p', the clear

line-of-sight C(p,p') may be computed by putting the clouds in order of increasing cloud cover:

]

= "cc -z% I)e- _ _,- 3

A3(l_ e- 3)+ 2(l-e ")+Al(1-e -x

=N3+ N 2

(6.17)

where AH<A j. In (6.17), we have dropped _ from x c/g for simplicity. As referenced to the upper

cloud group in Figure 3, the first and second brackets represent the absorption of the regions

where there are three and two cloud layers overlap, while the third bracket represents only the

absorption of the largest cloud cover.

It follows from (6.17) that for the case with J maximally overlapped clouds, C(p,p') may be

computed by evaluating the following recursion:

D _°J = 0 (6.18)

D o.) = Nj +D(J-l)e-ZCj/_t, j = 1..... J (6.19)

and

C(p,p') = 1-D °J (6.20)

Whenever a black cloud ('t_---_ oo) occurs in (6.19), all smaller clouds are eliminated from the

recursion. This treatment of cloud overlapping was first proposed by Chou and Suarez (1994). A

similar cloud overlapping scheme is also proposed by Raisanen (1998).

To provide an example for the overlapping scheme, we show in Figure 3 two groups of

maximally overlapped clouds between p and p'. The upper group and the lower group are labeled

U and L, respectively. These two groups are assumed randomly overlapped. The clear line-ofsight between p and p' is

C(p,p')=C v CL (6.21)

32
where C_ and CL are the clear line-of-sight of the upper and lower cloud groups computed from

(6.20). Fluxes can then be computed from (3.9) and (3.10) with T(p,p') replaced by C(p,p')T(p,p').

In the current version of our radiation code, clouds are grouped into three height ranges:

high, middle and low separated approximately by 400 hPa and 700 hPa pressure levels. Clouds

within each height group are assumed maximally overlapped, and clouds among the three height

groups are assumed randomly overlapped. This cloud overlapping scheme can be easily extended

to include a more flexible situation that assumes maximum overlapping of adjacent cloud layers

and random overlapping of cloud layers separated by clear layers, as originally proposed by

Gelygn and Hollingswoth (1979).

P

l

U

L

Figure 3. Schematic showing two groups of randomly overlapped clouds. Cloud layers within each group

are assumed maximally overlapped. CUand CLare the clear line-of-sight of the upper group and lower

group, respectively. The clear line-of-sight between the levels p and p' is CeCLas shown in (6.21).

7. AEROSOLS

The aerosol optical thickness, single-scattering albedo, and asymmetry factor for each of

the 10 bands and each of the atmospheric layers are specified input parameters. Different types of

aerosols are allowed to co-exist in a layer. The effective optical thickness, single-scattering

abledo, and asymmetry factor of a layer are computed similarly to that of clouds,

33
,_a = Z,ta i (7.1)

i

;a= Zo3a [_ai /Z_a i (7.2)

i i

ga=Zgatc.oai'cai/Zo3a{ta i (7.3)

i i

where i is the index for aerosol type. The same parameterization for cloud scattering, (6.11 ) and

(6.12), is applied to an aerosol-laden layer. We further assume that the aerosol layer is planeparallel and the flux transmission function of the layer is computed from

__:*a/_ T a = T e (7.4)

where "tTM is the scaled aerosol optical thickness according to (6.11) and (6.12), and 7_ and Tare

flux transmittances with and without aerosols.

8. VERTICAL DISCRETIZATION

8.1 Downward and Upward Fluxes

To approximate the vertical integrals in (3.9) and (3.10), the atmosphere is divided into L

layers numbered as shown in Figure 4. The downward and upward fluxes at level 1 for the ith

band are computed from

l-1

FiI$= Z B l[Ti(l,l'+l)-Ti(l,r)] ' ' i,l'+-

1'=1 2

1=2 ..... L+I (8.1)

L

Fi't'r= Z 8 _[T_(t,r) - T_(l,r+b ]

' i.l'+-

l'=l 2

+ Ti(I,L + 1)_,L+I T

1 = 1..... L (8.2)

34
T

Fl It' BI -112

BI + 112 T

TI, l"

1 1

BI'-I/2

BI '+1/2

1-1

/+1

1' +1

L

L+I

L+2

Figure 4. The vertical grid and placement of various quantities for an atmosphere consisting of L layers.

Quantities defined at the layers, such as the Planck flux, B are denoted by half-integer subscripts, and

quantities defined at the levels separating layers, such as the downward and upward fluxes /7, by integer

subscripts. The transmittance shown is for a multi-layer region bounded by levels l and l'. Note that the

surface is treated as a fictitious layer at L+3/2.

35
_,1 $ = 0 (8.3)

Fi.L+ I$ = Ei Bi (0s) + (1 - £i ) Fi,L+I $ (8.4)

where B_.j+_2 is the Planck flux at the temperature 0t.m integrated over the band i, Ti(l,l') is the

Planck-weighted flux transmittance between the levels l and l', T_(I,L+I) is the transmittance

between the level I and the Earth's surface, and 0s is the Earth's surface temperature.

By rearranging terms in Equations (8.1) and (8.2) and defining

AFiL (l,r) = Ti(l,r)l Bi ,l_,-g_l- Bi'l+_l )

Ah_?(I,/') = Ti(l,r) B I - B t I

i.l'-- 2 i,l' +_,)

BI=0

i,-

2

l' > l, l=l, 2..... L (8.5)

1' > 1, 1=1, 2 ..... L (8.6)

(8.7)

B 3 = esB_(0s)

i.L+-

2

the downward and upward fluxes can be written as

/'--I

FiJ'$= n,l'-I + Z AFi$(l'l')

2 l=l

Fi,/$ =B ¿ +Ti(I,L+I)(1-Ei)Fi,L+I $-

i,l +-

2

L+I

(8.8)

1'=2, 3..... L+I (8.9)

Z AFi$(I,I ') l =1, 2..... L (8.10)

/'=/+1

As opposed to (8.1) and (8.2), only one transmittance, T,4LI'), appears in each term under

the summation in (8.9) and (8.10). The sums can be build-up term by term. When the

transmittance of a layer bounded by the upper level I and the lower level 1' is computed, the

downward flux at 1' and upward flux at I are immediately undated. In this way, there is no need to

36
store the transmittance matrix. The storage of the entire routine scales then like L, rather than L 2

- a very significant advantage for models with high vertical resolution.

8.2 Treatment of Adjacent Layers

Accurate calculations of the contribution of a layer to the flux at the boundaries of the layer

are very important for two of reasons. (a) Among the atmospheric layers, the adjacent layers have

thc largc_t contribution to the fluxes. (b) The transmittance decreases rapidly within the adjacent

layer. Signilicant errors might be induced if the emitting temperature of the adjacent layer is

assumed Iobc thc mean layer temperature, as in (8.1) and (8.2). To reduce the error arising from

this as,umption, wc compute the emitting temperature (or Planck function) of the layer as a

function ol the layer transmittance and the temperatures at the upper and lower boundaries of the

laycr.

Thc contribution to the upward flux at the level p from the adjacent layer below can be

written as

+40

AF$(p) = B(p')[Or(p,p')/_p']dp'

(8.11)

= BU ( l- To )

where Ap is the thickness of the adjacent layer, /_' is the effective Planck flux of the adjacent

layer, and To = T(p + Ap, p) is the flux transmittance between p and p + Ap.

The Planck flux B(p') varies slowly within a layer, but the transmittance T(p, p')

decreases exponentially as p' departs from p. For To---_0, the adjacent layer is rather opaque, and

the radiation primarily comes from regions close to the level p. The effective Planck flux/V for

this case should be close to B(p). On the other hand, if the adjacent layer is rather transparent (To

---) 1), the transmitttance varies slowly with p' and the flux at p is contributed more or less evenly

from the entire layer. By linearizing the Plank flux within the layer, it follows that B_ _ B(p +

Ap/2) for To ---) 1. Calculations of the flux contribution from the adjacent layer should take into

account the dependence of B" on To.

Assuming that the Planck function varies linearly with pressure and the Planck-weighted

transmittance decreases exponentially with pressure away form p, we have B(p') = B(p) +

37
[B(p+z_qT) - B(p)] (p'-p) /Ap and T(p,p' )=e -c(p -P), where c is a constant. It follows from

(8.11 ) that the effective Planck flux of the adjacent layer is

=L l- To J L _ (8.12)

Equation (8.12) meets the conditions that B" -+B(p) for To -->0, and B u -[B(p + _)+ B(p)]/2

1

for weak absorption when To = 1-cAp+-_ (cap)"

Similarly, for the downward flux at the lower boundary of the layer, the effective Planck

function of the layer is

Bd = -B( p + 7-To )- B(p)T_J]+[LB( p + InTo Ap) - B(p) - (8.13)

From (8.12) and (8.13), we have

Bd + B u = B(p)+ B(p+ Ap)

By replacing Bi.r-os and Bi_+o5 with B a and B', Equations (8.9) and (8.10) reduce to

(8.14)

Fi,l'$=Bdi,l'-l+Ti(l',l'-l)ln ,_3-Bdi,l,_l__AFi$(l,l' ) (8.15)

/=1

F,.?'=8" ,-_(l.t+l)/S" ,-8 _]

L+I (8.16)

where

+ Ti(I'L + I)(I-Es)Fi'L+I$- Z Abi'l'(/'/')

1'=/+2

F

JE ]

l_.t - nirl'_ 1

B d l=lBirl'-t_fl'-lTi(l',l'-_ + ,

i,l'--_ L 1 - Ti(/"/'-l) . lnTi(t',t-1)

(8.17)

B u - (Bi, l +Bi, l')-B d (8.18) i,l 1- i,l,_ 1

2 2

and l'=l + l.

38
When a model layer spans a region where the temperature lapse rate changes signs, we

cannot assume that B linearly varies with p, and Equations (8.17) and (8.18) are no longer valid.

In such a case, we simply compute Ba and B" from

B u -B d 1 = 0.5B l+0.25(Bil+Bi,r) (8.19)

i,l 1- i.l'-- i.l +- '

2 2 2

8.3 The k-Distribution Transmission Functions

For gaseous absorption using the k-distribution method, the transmittance of each of the kintervals can be first computed for the L layers from

AT -k, Aw;+v2/_ 1 = 1, 2..... L; n = 1, 2..... N (8.20) 1--e

n,l +-

2

where k, is the absorption coefficient of the n 'h k-interval, and Aw;÷;/2_is the scaled absorber

amount of the layer between the levels I and l+l. Both k and Aw vary with the spectral interval i.

This index is dropped for simplici_. Because the set of k's are chosen according to (4.12), we

need only to compute the exponential for the first value of k, and the others can be derived from

ATn = (ATn-I) q n = 2, 3 ..... U (8.21)

Once AT_ is computed, the computation of each of the AT, with n>l requires only a few

multiplication operations.

The transmittance between an upper level I and an lower level l', i.e. l '> l, is given by

L; l' =1+1 ..... L+I (8.22)

t-I

j=l

and the transmittance between the level I and the surface is given by

Tn(I,L+I)=NATn,jl

j=l

(8.23)

Following (4.10), the band-averaged transmission terms in (8.5) and (8.6) are computed from

39
N

T( I,I') = E Tn( l,l')Agn

n=l

8.4 Table Look-Up for Transmission Functions

When using the pre-computed transmittance tables, the transmittance between the upper

level / and the lower level l' is derived from

Tim = TCul.r,7,t.r,0ta,)

(8.24)

where u is the Cnon-scaled) absorber amount, and _ and 0 are the effective pressure and

temperature. They are defined by

I'--]

Ul.t "= E Attj+I

j=l 2

(8.25)

(8.26)

I'--I

EP l Au . 1

j=t g_ g+-2

pl .r = /'-l (8.27)

EAu 1

1'--1

"_0 iAu 1

ot.r j=t

- "_'-i (8.28)

where j+ 1/2 denotes the layer between the levels j and j+ 1. In (8.25), the transmittance T varies

with the spectral band i. This index is dropped for simplicity.

4O
9. SUB-GRID VARIATIONS OF LAND SURFACE

If the size of a model grid-box is greater than tens of kilometers, the variations of surface

temperature and emissivity within the grid box could have a significant effect on model

simulations. In such a case, the emissivity and reflectivity of the model surface should include the

effect of the sub-grid variations. Let us consider a vegetation-covered land surface characterized

by a ground-skin temperature 0g, vegetation canopy temperature 0v, ground emissivity _g, canopy

emissivity Ev, and canopy reflectivity r_. The radiation emitted by the ground and the vegetation

that leaves the top of the canopy is given by

Tv[EgB(Og) + rgEvB(Ov )]

E= +evB(Ov) (9.1)

(1- rvrg)

where r_ = (1-E_) is the ground reflectivity, and Tv = (l-e,,-r,.) is the canopy transmissivity. The

first term within the bracket is the radiation emitted by the ground, the second term within the

bracket is the radiation emitted downward by the vegetation and subsequently reflected by the

ground, and the last term is the upward emission by the vegetation. The denominator in (9.1)

takes care of the multiple reflections between the vegetation and the ground.

The downward flux from the atmosphere is reflected by the vegetation and the ground. This

term is given by

R = rl_ (Ps) (9.2)

where r is the effective reflectivity of the canopy and the ground given by

r rv2

r = rv + (9.3)

(1- rvrg)

For the case without vegetation cover, we have Tv = 1, v_ = rv = 0, and Equations (9.1) and

(9.3) reduce to

E=EgB(Og) (9.4)

r = rg (9.5)

It should be noted that Equations (3.10), (8.2) and (8.4) are derived based on (9.4) and (9.5).

41
If there are N sub-grid boxes in a model box with fractional cover f,, n =

Equations (9. I ) and (9.3) become

1_ 2_ °.. N_

N

E= Zrf_

n=l

(9.6)

r = Fv, n

n=l

(9.7)

where

N

Z f. =1

n=l

(9.8)

and the subscript n denotes the sub-grid box.

For the case without vegetation cover, Equations (9.6) and (9.7) reduce to

N

E = £ fn£g, nB(Og, n )

n=l

(9.9)

N

r = Lf n rg,n

n=l

(9.10)

With E and r given above and R given by (9.2), Equations (8.2) and (8.4) reduce to the following

general forms

L

b)$ = Z B,+/[T( 1,Z') - T(l,l'+D ]+ T(l, L + 1)FL+I "1" (9.1 l)

2 l'=l

FL+I $ = E+ R= E+ rFL+I $ (9.12)

Finally, the partial derivative of the upward flux with respect to the surface temperature given by

(3.12) reduces to the following general forms

42
F/=|

(9.13)

_S

- T(p,ps)E' (9.14)

where B'(O) is the derivative of B with respect to temperature at 0. For simplicity, we have

omitted the index for the spectral band i in all of the equations.

43
10. COMPARISONS WITH LINE-BY-LINE CALCULATIONS

Fluxes and cooling rate computed using the transmittance parameterizations are compared

with high spectral-resolution line-by-line calculations. Our line-by-line calculations of the

absorption coefficient use the Air Force Geophysical Laboratory 1996 edition of the molecular

absorption parameters (Rothman et al., 19981). The molecular line shape is assumed to follow the

Voigt function. The absorption coefficient at wavenumbers > 10 cm -1 from the line center is

taken to be zero, which is equivalent to a line-cutoff of 10 cm j. The absorption coefficient is

computed at spectral intervals of 0.00t cm -_for Band 3 and 0.002 cm -j for the other bands.

The mid-latitude summer atmosphere and the sub-arctic winter atmosphere taken from

McClatchey et al. (1972) are used in the flux calculations. No clouds and aerosols are included.

The C02 concentration is fixed at 350 ppmv, and the specific humidity above the tropopause is set

to 4x10 6 g/g. The atmosphere is divided into 75 layers with Ap - 25 hPa at pressures > 100 hPa

and Alogjd 9 = 0.15 at pressures < 100 hPa.

Cooling rates computed using the line-by-line method (solid lines) and the parameterization

with the HIGH option (dashed lines) are shown in Figures 5 and 6. The cooling rate is presented

as a function of log_op in the upper panels and p in the lower panels. It is noted that for the HIGH

option, the transmission functions of C02 in Band 3 and of water vapor in Bands 1, 2, and 8 are

computed using table look-up (see Table 1). It can be seen in the figures that, compared to the

line-by-line method, the cooling rate is computed accurately. The maximum error in the

stratosphere is < 0.25 °C/day and is < 0.15 °C/day in the troposphere. These errors are small when

compared with the maximum cooling of -12 °C/day in the stratosphere and -2°C/day in the

troposphere.

The downward flux at the surface and the upward flux at the top of the atmosphere are

shown in Tables 14 and 15 for the mid-latitude summer atmosphere and the sub-arctic winter

atmosphere, respectively. Errors are generally smaller than 1 Wm -2 in individual bands.

Integrated over the LW spectrum, the parameterizations can compute the fluxes to within 1% of

the line-by-line calculations for the mid-latitude summer atmosphere and within 0.5% for the subarctic winter atmosphere.

44
O..

t---

v

o_

t3_

r_

¢.-

v

(:1.

I 0 -2

10 -1

lo0

102

103

0

2OO

400

6O0

8OO

1000

0.00.S

' ,_i • ' ' i • • ' i • • •

"--_ Line-By -kine - - Parameterization

,"_,, , , , I , , , I , , , I , , , i , . , I . . ,

0 2 4 6 8 10 12 14

............ If-: .... ;--- , --.

.,.,I .... i ....

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Cooling Rate (C/day)

Figure 5. The cooling rate computed for a clear mid-latitude summer atmosphere using the line-by-line

method (solid curves) and the "HIGH" option of the parameterization (dashed curves). Cooling is due to

water vapor molecular line and continuum absorption, as well as C02 and 03 absorption.

45
{3_

J::

l/3

{:L

{3_

¢.-

-I

(D

(/I

o

{3_

10.2

I0 "1

10°

101

102

103

0

200

400

600

800

1000

0 2 4 6 8 10 12 14

.... i .... i" ,,i -, ....

0.0

'''I'''I '''I'''I ''' I'''I'''Ii_ ..... Line-By-Line

k,-.f, I , • ,l • • , I ,* • I * ,, I , , ,I , , , I ,, ,

16

0.5 1.0 1.5 2.0 2.5

Cooling Rate (C/day)

Figure 6. Same as Figure 5 except for a clear sub-arctic winter atmosphere.

46
Table14.Downwardfluxesatthesurface,F+s1,.,and upward fluxes at the top, F_,op, for

a clear mid-latitude summer atmosphere computed using a line-by-line method and the

parameterization with the HIGH option. Units of the fluxes are Win-:.

Spectral Band (cm -_) F+,¢, F * Absorber

• top

0- 340

line-by-line

Parameterization

34O- 540

line-by-line

Parameterization

540- 800

line-by-line

Parametenzation

800- 980

line-by-line

Parametenzation

980-1100

line-by-line

Parameterization*

1100-1215

line-by-line

Parametenzation

1215-1380

line-by-line

Parametenzation

1380-1900

line-by-line

Parameterization

1900-3000

line-by-line

Parametenzation

Total

line-by-line

Parameterization

51.02 34.56 H ,O (line)

51.04 34.40 11,_0 (line)

80.76 61.01 H20 (line)

81.23 60.54 H20 (line)

105.85 67.90 H20 (line, continuum), C02

107.29 68.29 H20 (line, continuum), C02

26.89 58.62 H20 (line, continuum), O_

27.01 58.72 H20 (line, continuum)

12.36 22.36 H20 (line, continuum), Oj

12.54 21.41 H20 (line, continuum), O:

9.73 20.96 H20 (line, continuum), 03

9.63 21.42 H20 (line, continuum)

19.43 15.89 H20 (line, continuum)

19.78 15.67 H20 (line, continuum)

30.51 7.56 1-120 (line)

30.68 7.61 1-120 (line)

3.04 5.13 H _O (line)

3.17 5.00 H20 (line)

339.59 293.99

342.37 293.03

* The absorption due to 03 in Bands 4 and 6 are folded into the 03 absorption tables of this

band, see Equation (4.18).

47
Table15.SameasTable14,exceptforaclearsub-arcticwinteratmosphere.

SpectralBand(cm -1) F+sjc F _ Absorber

top

0- 340

line-by-line

Parameterization

340- 540

line-by-line

Parametenzation

540- 800

line-by-line

Parametenzation

800- 980

line-by-line

Parameterization

980-1100

line-by-line

Parameterization*

1100-1215

line-by-line

Parameterization

1215-1380

line-by-line

Parametenzation

1380-1900

line-by-line

Parametenzation

1900-3000

line-by-line

Parameterization

Total

line-by-line

Parameterization

40.35 32.21 H,.O (line)

40.25 32.08 1-120 (line)

47.34 52.20 H20 (line)

47.16 51.97 H20 (line)

53.54 51.13 H20 (line, continuum), C02

51.73 51.40 H20 (line, continuum), C02

1.46 32.85 HzO (line, continuum), O_

!.61 32.84 H20 (line, continuum)

3.13 11.01 1120 (line, continuum), Oj

3.37 10.59 H20 (line, continuum), 05

1.06 9.97 H20 (line, continuum), 03

0.97 10.25 H,.O (line, continuum)

4.61 8.62 11,.0 (line, continuum)

4.86 8.60 H20 (line, continuum)

10.10 5.01 H20 (line)

10.19 5.09 H20 (line)

0.42 1.37 H20 (line)

0.44 !.38 1-120 (line)

162.01 204.37

161.09 204.21

* The absorption due to O_ in Bands 4 and 6 are folded into the 03 absorption tables of this

band, see Equation (4.18).

48
Thereareatotalof 13minorabsorptionbandsincludedin theLW code (see Table 1).

Thesc absorption bands are important in studying the greenhouse climatic effect due to human

activities. Table 16 shows the effect of these bands on the fluxes at the top of the atmosphere and

thc surfacc. It is the difference between the flux with all the minor bands included and the flux

with thc absorption due to a given trace gas excluded. The concentration of these trace gases used

is ().21_ ppmv for NO, 1.75 ppmv for CH4, 0.3 ppbv for CFCI 1, 0.5 ppbv for CFC12, 0.2 ppbv for

CFC22. and 350 ppbv for C02. The effect of these bands is a reduction of the outgoing longwave

radiation at the TOA by 2.5 - 5.0 Wm 2 and an enhancement of the downward longwave radiation

at the surface by-3 Wm 2.

Table 16. Effects of the minor absorption bands on the fluxes at the top of the

atmo,phcrc AFT,,,p and the surface AF_src as calculated using the LW radiation

parametcrization. See the text for the gas concentrations used in flux calculations. Units

of the Iquxcs are Wm -_'.

CH 4 N,O CFC's Minor C02 Total

Mid-Latitude Summer

AFt,or -2.22 - ! .83 -0.47 -0.38 -4.91

AF_qc +0.86 +0.58 +0.39 +0.86 +2.68

Sub-Arctic Winter

AF_,op -1.00 - 1.10 -0.19 -0.07 -2.36

AF_¢,, + 1.21 + 1.23 +0.39 +0.37 +3.20

49
11. CONCLUDING REMARKS

This technical memorandum documents the LW radiation code developed at the Climate

and Radiation Branch, NASA/Goddard Space Flight Center, which has been implemented in a

number of cloud, weather, and climate models. This code includes all the major absorbers and

most of the minor absorbers. Cloud optical properties are computed as functions of the ice/liquid

water content, the effective particle size, and spectral bands. Aerosol optical properties are

specified as input to the code. Scattering due to clouds and aerosols is included by scaling the

optical thickness. To enhance the accuracy and speed of flux calculations, different approaches

have been applied to different spectral bands for transmission calculations. In the vertical

integration of fluxes, the adjacent layers to a level where fluxes are computed are treated

separately in order to achieve a high degree of accuracy. Sample calculations show that this code

can compute accurately fluxes and cooling rate within 1% and 0.15 C/day (0.25 °C/day in the

stratosphere), respectively, of the line-by-line calculations.

Depending upon the purpose of the LW flux calculations and the nature of the atmospheric

model, there are a number of options available for efficiently running this code;

When the upper stratosphere is not crucial in a study, one can choose the option high=.false.

In this case, cooling rate in the upper stratosphere above the lO-hPa level is greatly

underestimated. When one chooses high=.true., the cooling rate is computed accurately from

the surface to the O.Ol-hPa level, but the computation is slower.

. When the absorption in the minor bands (N_O, CH4, CFC's, and minor C02) are not important

in certain studies, one can choose trace=.false. The absorption due to those bands is not

included.

,

.

When a vegetation layer is added, the emission and reflection of both the ground and the

canopy are included in computing the radiation emitted and reflected by the combined

vegetation-ground surface. Set vege=.true, if a vegetation layer is included.

For many cloud and meso-scale models, the size of a grid box is small. Factional cloud cover

of a layer is either 0 or 1. Treatment of overlapping of clouds at different layers is not an

issue, and the option overcast=.true, can be used to reduce computing time.

50
, The cloud optical thickness either can be computed from the cloud ice/liquid water content

and the effective mean particle size or can be an input to the LW code. Set cldwater=.true, for

the former and cldwater=.false, for the latter,

6. Set aerosol=.true, if aerosols are included in computing atmospheric transmission functions.

Otherwise set aerosol=.false.

There are still rooms for improvement to the code especially the parameterization for cloud

optical thickness and the overlapping among cloud layers. To be available for use by atmospheric

researchers, we have put this code and the documentation at

http://climate.gsfc.nasa.gov/-chou/clirad_lw. They will be updated with new developments of the

parameterization.

ACKNOWLEDGMENT

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

Office of Earth Science, NASA Headquarters.

51
REFERENCES

Arking, A., and K. Grossman, 1972: The influence of line shape and band structure on

temperatures in planetary atmospheres. J. Atmos. Sci., 29, 937-949.

Bacmeister. J. T.. and M. J. Suarez,, 2001: Wind-stress simulations and equatorial dynamics in an

AGCM. Part I: Basic results from a 1979-1999 forced SST experiment. J. Atmos. Sci., 58.

Chou. M.-D.. and A. Arking, 1980: Computation of Infrared Cooling Rates in the Water Vapor

Bands. J. Atmos. Sci., 37, 855-867.

Chou. M.-D.. and L. Kouvaris, 1991: Calculations of Transmission Functions in the IR C02 and

O. Band_. J. Geophys. Res., 96, 9003-9012.

Chou, M.-I). and M. Suarez, 1994: An efficient thermal infrared radiation parameterization for

use in general circulation models. NASA Tech. Mem. 104606, 3, Technical Report Series on

Global Modcling and Data Assimilation. 85 pp.

Chou, M.-D., W. Ridgway, and M.-H. Yan, 1993: One-Parameter Scaling and Exponential-Sum

Fitting for Water Vapor and CO, Infrared Transmission Functions. J. Atmos. Sci., 50, 2294-

2303.

Chou, M.-D., K.-T. Lee, S.-C. Tsay, and Q. Fu, 1999: Parameterization for cloud longwave scattering for

use in atmospheric models. J. Climate, 12, 159-169.

Chou, M.-D., M. J. Suarez, C. H. Ho, M. M. H. Yan, and K. T. Lee, 1998: Parameterizations for

cloud overlapping and shortwave single-scattering properties for use in general circulation

and cloud ensemble models. J. Climate, 11,202-214.

Clough, S. A., F. X. Kneizys, and R. W. Davies, 1989: Line shape and the water vapor

continuum. Atmos. Res., 23, 229-241.

Fu, Q, and K.-N. Liou, 1992: On the correlated k-distribution method for radiative transfer in

nonhomogeneous atmospheres. J. Atmos. Sci., 49, 2153-2170.

Fu, Q., P. Yang, and W. B. Sun, 1998: An accurate parameterization of the infrared radiative

properties of cirrus clouds for climate models. J. Climate, 11, 2223-2237.

52
Geleyn, J.-F., and A. Hotlingsworth, 1979: An economical analytical method for the computation

of the interaction between scattering and line absorption of radiation. Beitr. Phys. Atmos.,

52, 1-16.

Goody, R. M., R. West, L. Chen, and D. Crisp, 1989: The correlated-k method for radiation

calculation in nonhomogeneous atmospheres. J. Quant. Spectrosc. Radiat. Transfer, 42,

539-550.

Harshvardhan, D. A. Randall and T. G. Corsetti, 1987: A fast radiation parameterization for

atmospheric circulation models. J. Geophy. Res., 92, 1009-1016.

Kratz, D. P., M.-D. Chou, and M. M.-Y. Yan, 1993: Infrared radiation parameterizations for the

minor CO,. bands and for several CFC bands in the window region. J. Climate, 6, 1269-1281.

Kratz, D. P., M.-D. Chou, M. M.-H. Yan, and C.-H. Ho, 1998: The k-distribution method for

computing climate radiative forcing due to minor trace gases. J. Geophy. Res., 103, 31647-

31656.

Lacis, A., and V. Oinas, 1991: A description of the correlated k-distribution method for modeling

nongray gaseous absorption, thermal emission, and multiple scattering in vertically

inhomogeneous atmospheres. J. Geophy. Res., 96, 9027-9063.

Li, J., 2000: Accounting for overlap of fractional cloud in infrared radiation. Q. J. R. Meteor.

Soc., 126, 3325-3342.

Liang, X.-Z., W.-C. Wang, 1997: Cloud overlap effects on general circulation model climate

simulations. J. Geophy. Res., 102, 11039-11047.

Manabe, S., and R. E. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective

adjustment. J. Atmos. Sci., 21,361-385.

McClatchey, R. A., R. W. Fenn, J. E. A. Selby, F. E. Volz and J. S. Gating, 1972: Optical

properties of the atmosphere, 3rd ed. AFCRL-72-0497, 108 pp.

Mlawer, E. J., S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, 1997:

Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k

model for the longwave. J. Geophy. Res., 102, 16663-16682.

53
Raisanen, P.,1998:Effectivelongwavecloudfractionandmaximum-randomoverlapof clouds:

A problemandasolution.Mon. Wea. Rev., 126, 3336-3340.

Roberts, R. E., J. E. A. Selby and L. M. Biberman, 1976:: Infrared continuum absorption by

atmospheric water vapor in the 8-12 iam window. Appl. Opt., 15, 2085-2090.

Rothman, L. S., and co-authors, 1998: The HITRAN molecular spectroscopic database and

HAWK (HITRAN Atmospheric Workstation): 1996 Edition. J. Quant. Spectrosc. Radiat.

Transfer, 60, 665-7 !0.

Sud, Y., and D. M. Mocko, 1999: New snow-physics to complement SSiB. Part I: Design and

evaluation with ISLSCP initiative I datasets. J. Meteor. Soc. Japan, 77, 335-348.

Tao, W.-K., J. Simpson, D. Baker, S. Braun, M.-D. Chou, B. Ferrier, D. Johnson, A. Khain, S.

Lang, B. Lynn, C.-L. Shie, D. Starr, C.-H. Sui, Y. Wang and P. Wetzel, 2001: Microphysics,

radiation and surface processes in a non-hydrostatic model, Meteor. and Atmos. Phys.

(submitted).

Wang, W.-C., and G. Y. Shi, 1988: Total band absorptance and k-distribution function for

atmospheric gases. J. Quant. Spectrosc. Radiat. Transfer, 39, 387-397.

Wu, M.-L. C., 1980: The exchange of infrared energy in the troposphere. J. Geophy. Res., 85,

4084-4090.

54
Previous Volumes in This Series

Volume 1

September 1994

Volume 2

October 1994

Volume 3

December 1994

Volume 4

January 1995

Volume 5

April 1995

Volume 6

April 1995

Volume 7

September 1995

Volume 8

March 1996

Documentation of the Goddard Earth Observing System

(GEOS) general circulation model - Version 1

L.L. Takacs, A. Molod, and T. Wang

Direct solution of the implicit formulation of fourth order

horizontal diffusion for gridpoint models on the sphere

Y. Li, S. Moorthi, and J.R. Bates

An efficient thermal infrared radiation parameterization for

use in general circulation models

M.-D. Chou and M.J. Suarez

Documentation of the Goddard Earth Observing System

(GEOS) Data Assimilation System - Version 1

James Pfaendtner, Stephen Bloom, David Lamich,

Michael Seablom, Meta Sienkiewicz, James Stobie,

and Arlindo da Silva

Documentation of the Aries-GEOS dynamical core: Version 2

Max J. Suarez and Lawrence L. Takacs

A Multiyear Assimilation with the GEOS-1 System:

Overview and Results

Siegfried Schubert, Chung-Kyu Park, Chung-Yu Wu,

Wayne Higgins, Yelena Kondratyeva, Andrea Molod,

Lawrence Takacs, Michael Seablom, and Richard

Rood

Proceedings of the Workshop on the GEOS-1

Assimilation

Siegfried D. Schubert and Richard B. Rood

Five-Year

Documentation of the Tangent Linear Model and Its Adjoint

of the Adiabatic Version of the NASA GEOS-1 C-Grid GCM:

Version 5.2

Weiyu Yang and I. Michael Navon

55
Volume 9

March 1996

Volume 10

April 1996

Volume 11

May 1997

Volume 12

August 1997

Volume 13

December 1997

Volume 14

August 1998

Volume 15

June 1999

Volume 16

November 1999

Volume 17

Jury ooo

Volume 18

December 2000

Energy and Water Balance Calculations in the Mosaic LSM

Randal D. Koster and Max J. Suarez

Dynamical Aspects of Climate Simulations Using the GEOS

General Circulation Model

Lawrence L. Talmcs and Max J. Suarez

Documentation of the Tangent Linear and its Adjoint Models

of the Relaxed Arakawa-Schubert Moisture Parameterization

Package of the NASA GEOS-1 GCM (Version 5.2)

Weiyu Yang I. Michael Navon, and Ricardo Todling

Comparison of Satellite Global Rainfall Algorithms

Alfred T.C. Chang and Long S. Chiu

Interannual Variability and Potential Predictability in Reanalysis Products

Wie Ming and Siegfried D. Schubert

A Comparison of GEOS Assimilated Data with FIFE

Observations

Michael G. Bosilovich and Siegfried D. Schubert

A Solar Radiation Parameterization for Atmospheric Studies

Ming-Dah Chou and Max J. Suarez

Filtering Techniques on a Stretched Grid General Circulation

Model

Lawrence Takacs, William Sawyer, Max J. Suarez,

and Michael S. Fox-Rabinowitz

Atlas of Seasonal Means Simulated by the NSIPP-1 Atmospheric GCM

Julio T. Bacmeister, Philip J. Pegion, Siegfried D.

Schubert, and Max J. Suarez

An Assessment of the Predictability of Northern Winter Seasonal Means with the NSIPP 1 AGCM

Philip J. Pegion, Siegfried D. Schubert, and Max J.

Suarez

56
w _
REPORT DOCUMENTATION PAGE FormApproved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources.

gathering and maintaining the data needed, and completing and reviewing the collection Of intormalion. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for In|ormation Operations and Reports. 1215 Jefferson

Davis Highway. Suite 1204, Arlington. VA 22202-4302. and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188). Washington. DC 20503.

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

July 2001 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Technical Report Series on Global Modeling and Data Assimilation

Volume 19--A Thermal Infrared Radiation Parameterization for

Atmospheric Studies Code 971

6. AUTHOR(S)

Ming-Dah Chou, Max J. Suarez, Xin-Zhong Liang, and Michael M.-H. Yan

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

Goddard Space Flight Center

Greenbelt, Maryland 20771

9.SPONSORINGl MONITORINGAGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PEFORMING ORGANIZATION

REPORT NUMBER

2001-03197-0

10. SPONSORING I MONITORING

AGENCY REPORT NUMBER

TM-2001-104606, Vol. 19

11. SUPPLEMENTARY NOTES

X.-Z. Liang: University of Illinois at Urbana-Champaign,Champaign, Illinois

M. M.-H. Yan: Sciences Systems and Applications, Inc., Lanham, Maryland

12a. DISTRIBUTION I AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified-Unlimited

Subject Category: 46

Report available from the NASA Center for AeroSpace Information,

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

13. ABSTRACT (Maximum 200 words)

This technical memorandum documents the longwave radiation parameterization developed at the Climate and

Radiation Branch, NASA/Goddard Space Flight Center, for a wide variety of weather and climate applications.

Based on the 1996-version of the Air Force Geophysical Laboratory HITRAN data, the parameterization

includes the absorption due to major gaseous absorption (water vapor, CO,, O_) and most of the minor trace

gases (N20, CH4, CFC's), as well as clouds and aerosols. The thermal infrared spectrum is divided into nine

bands. To achieve a high degree of accuracy and speed, various approaches of computing the transmission

function are applied to different spectral bands and gases. The gaseous transmission function is computed either

using the k-distribution method or the table look-up method. To include the effect of scattering due to clouds

and aerosols, the optical thickness is scaled by the single-scattering albedo and asymmetry factor. The parameterization can accurately compute fluxes to within 1% of the high spectral-resolution line-by-line calculations.

The cooling rate can be accurately computed in the region extending from the surface to the 0.01-hPa level.

14. SUBJECT TERMS

Thermal Infrared Radiation, Atmospheric Models, k-Distribution Method,

Minor Trace Gases

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

56

16. PRICE CODE

20. LIMITATION OF ABSTRACT=

UL

Standard Form 298 (Rev. 2-89)