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4.9 Parameterization of Longwave Radiation

The method employed in the CAM 3.0 to represent longwave radiative transfer is based on an absorptivity/emissivity formulation [138]
$\displaystyle F^\downarrow (p)$ $\displaystyle =$ $\displaystyle B(p_t)\varepsilon (p_t,p)+ \int\limits_{p_t}^p {\alpha
(p,p')dB(p')}$ (4.229)
$\displaystyle F^\uparrow
(p)$ $\displaystyle =$ $\displaystyle B(p_s)-\int\limits_p^{p_s} {\alpha (p,p')dB(p')} ,$ (4.230)

where $ B(p)=\sigma T(p)^4$ is the Stefan-Boltzmann relation. The pressures $ p_t$ and $ p_s$ refer to the top of the model and the surface, respectively. $ \alpha$ and $ \epsilon$ are the absorptivity and emissivity
$\displaystyle \alpha (p,p')$ $\displaystyle =$ $\displaystyle \frac{\int\limits_0^\infty { \left\{ dB_\nu (p')
/dT(p') \right\} ( 1-{\cal T}_\nu (p,p')) \; d\nu}}{dB(p)/dT(p)}$ (4.231)
$\displaystyle \varepsilon (p_t,p )$ $\displaystyle =$ $\displaystyle \frac{\int\limits_0^\infty {B_\nu (p_t) ( 1-{\cal T}_\nu (p_t,p)) \; d\nu}}{B(p_t)},$ (4.232)

where the integration is over wavenumber $ \nu $. $ B_\nu(p) = B_\nu(T(p))$ is the Planck function, and $ {\cal T}_\nu $ is the atmospheric transmission. Thus, to solve for fluxes at each model layer we need solutions to the following:

$\displaystyle \int\limits_0^\infty {\left( {1-{\cal T}_\nu } \right)}F(B_\nu )d\nu,$ (4.233)

where $ F(B_\nu )$ is the Planck function for the emissivity, or the derivative of the Planck function with respect to temperature for the absorptivity.

The general method employed for the solution of (4.233) for a given gas is based on the broad band model approach described by Kiehl and Briegleb [87] and Kiehl and Ramanathan [93]. This approach is based on the earlier work of Ramanathan [136]. The broad band approach assumes that the spectral range of absorption by a gas is limited to a relatively small range in wavenumber $ \nu $, and hence can be evaluated at the band center, i.e.

$\displaystyle \int_{\nu_1}^{\nu_2} {(1-{\cal T}_\nu )F\left( {B_\nu } \right)d\...
...\nu }} \right)\int_{\nu_1}^{\nu_2} {(1-{\cal T}_\nu )d\nu = F(B_{\bar\nu})A }},$ (4.234)

where $ A$ is the band absorptance (or equivalent width) in units of cm$ ^{-1}$. Note that $ A$, in general, is a function of the absorber amount, the local emitting temperature, and the pressure. Thus, the broad band model is based on finding analytic expressions for the band absorptance. Ramanathan [136] proposed the following functional form for $ A$:

$\displaystyle A(u,T,P)=2A_{0} \ln \left\{ 1+\frac{u}{\sqrt {4+u(1+1/\beta)}} \right\},$ (4.235)

where $ A_0$ is an empirical constant. $ u$ is the scaled dimensionless path length

$\displaystyle u=\int {\frac{S(T)}{A_0(T)}\mu\rho _adz},$ (4.236)

where $ S(T)$ is the band strength, $ \mu$ is the mass mixing ratio of the absorber, and $ \rho_a$ is the density of air. $ \beta$ is a line width factor,

$\displaystyle \beta = \frac{4}{ud}\int {\gamma (T)\left( \frac{P}{P_0} \right) du},$ (4.237)

where $ \gamma(T)$ is the mean line halfwidth for the band, $ P$ is the atmospheric pressure, $ P_0$ is a reference pressure, and $ d$ is the mean line spacing for the band. The determination of $ \gamma$, $ d$, $ S$ from spectroscopic line databases, such as the FASCODE database, is described in detail in Kiehl and Ramanathan [93]. Kiehl and Briegleb [87] describe how (4.235) can be extended to account for sub-bands within a spectral region. Essentially, the argument in the log function is replaced by a summation over the sub-bands. This broad band formalism is employed for CO$ _2$, O$ _3$, CH$ _4$, N$ _2$O, and minor absorption bands of CO$ _2$, while for the CFCs and stratospheric aerosols we employ the exponential transmission approximation discussed by Ramanathan et al. [139]

$\displaystyle T=\exp \left[ - D \left( S(T) / \Delta\nu \right) W \right],$ (4.238)

where $ \Delta\nu$ is the band width, and $ W$ is the absorber path length

$\displaystyle W=\int {\mu\rho _adz},$ (4.239)

and $ D$ is a diffusivity factor. The final problem that must be incorporated into the broad band method is the overlap of one or more absorbers within the same spectral region. Thus, for the wavenumber range of interest, namely 500 to 1500 cm$ ^{-1}$, the radiative flux is determined in part by the integral

$\displaystyle \int_{500}^{1500} { (1-{\cal T}_\nu)F(B_\nu)d\nu },$ (4.240)

which can be re-formulated for given sub intervals in wavenumber as
$\displaystyle \int_{500}^{1500} { (1-{\cal T}_\nu)F(B_\nu) d\nu} =
...{CO_2}^1{\cal T}_{N_2O}^1{\cal
T}_{H_2O}{\cal T}^{1}_{H_2SO_4}) F(B_\nu) d\nu }$      
$\displaystyle +\int_{750}^{820} {
(1-{\cal T}_{CFC11}^1{\cal T}_{H_2O}{\cal T}^{*}_{H_2SO_4})F(B_\nu) d\nu} \;$      
$\displaystyle +
\int_{820}^{880} { (1-{\cal T}_{CFC11}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}) F(B_\nu)
$\displaystyle + \int_{880}^{900} { (1-{\cal T}_{CFC12}^1{\cal
T}_{H_2O}{\cal T}^{3}_{H_2SO_4})F(B_\nu) d\nu} \;$      
$\displaystyle + \int_{900}^{1000} {(1-{\cal
T}_{CO_2}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\cal T}_{CFC11}^3 {\cal
$\displaystyle + \int_{1000}^{1120} {
(1-{\cal T}_{CO_2}^3{\cal T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4} {\cal
T}_{CFC11}^4{\cal T}_{CFC12}^3)F(B_\nu) d\nu}$      
$\displaystyle +
\int_{1120}^{1170} { (1-{\cal T}_{CFC12}^4{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}{\cal
T}_{N_2O}^2) F(B_\nu) d\nu} \;$      
$\displaystyle + \int_{1170}^{1500} { (1-{\cal
T}_{CH_4}{\cal T}_{N_2O}^3{\cal T}_{H_2O}{\cal T}^{5}_{H_2SO_4}) F(B_\nu) d\nu}$     (4.241)

The factors $ {\cal T}^{i}_{H_2SO_4}$ represent the transmissions through stratospheric volcanic aerosols. The transmissions in each band are replaced by effective transmissions $ {\bar T}^{i}_{H_2SO_4}$ given by:

$\displaystyle {\bar T}^{i}_{H_2SO_4} = \exp\left(-D \kappa_{i,volc} W_{volc}\right)$ (4.242)

where $ D = 1.66$ is the diffusivity factor, $ \kappa_{i,volc}$ is an effective specific extinction for the band, and $ W_{volc}$ is the mass path of the volcanic aerosols. For computing overlap with minor absorbers, methane, and carbon dioxide, the volcanic extinctions are computed for five wavenumber intervals given in table 4.2. The transmissions for overlap with the broadband absorption by water vapor are defined in equation 4.275. The volcanic transmission for the 798 cm$ {}^{-1}$ band of N$ {}_2$O is

$\displaystyle {\bar T}^{*}_{H_2SO_4} = 0.7 {\bar T}^{2}_{H_2SO_4} + 0.3 {\bar T}^{3}_{H_2SO_4}$ (4.243)

Table 4.2: Wavenumber Intervals for Volcanic Specific Extinctions
Index $ \nu_1-\nu_2$
1 500 - 650
2 650 - 800
3 800 - 1000
4 1000 - 1200
5 1200 - 2000

The sub-intervals in equation 4.241, in turn, can be reformulated in terms of the absorptance for a given gas and the ``overlap'' transmission factors that multiply this transmission. Note that in the broad band formulation there is an explicit assumption that these two are uncorrelated (see Kiehl and Ramanathan [93]). The specific parameterizations for each of these sub-intervals depends on spectroscopic data particular to a given gas and absorption band for that absorber.

4.9.1 Major absorbers

Details of the parameterization for the three major absorbers, H$ _2$O, CO$ _2$ and O$ _3$, are given in Collins et al. [40], Kiehl and Briegleb [87], and Ramanathan and Dickinson [137], respectively. Therefore, we only provide a brief description of how these gases are treated in the CAM 3.0. Note that the original parameterization for H$ _2$O by Ramanathan and Downey [138] has been replaced a new formulation in CAM 3.0.

For CO$ _2$

$\displaystyle \alpha_{CO_2}(p,p') = \frac{1}{4 \sigma T^3(p')} \frac{dB_{CO_2}}{dT'} (p') A_{CO_2} (p',p).$ (4.244)

$ B_{CO_2}$ is evaluated for $ \tilde{\nu} = 667$ cm$ ^{-1}$, where $ A_{CO_2} (p',p)$ is the broad-band absorptance from Kiehl and Briegleb [87]. Similarly,

$\displaystyle \epsilon_{CO_2} (0,p) = \frac{1}{\sigma T^4(0)} B_{CO_2} (0) A_{CO_2} (0,p).$ (4.245)

For ozone,

$\displaystyle \alpha_{O_3}(p, p')$ $\displaystyle = \frac{1}{4 \sigma T^3(p')} \frac{dB_{O_3}}{dT'} (p') A_{O_3} (p',p),$ (4.246)


$\displaystyle \epsilon_{O_3} (0,p)$ $\displaystyle = \frac{1}{\sigma T^4(0)} B_{O_3} (0) A_{O_3} (0,p),$ (4.247)

where $ A_{O_3}$ is the ozone broad-band absorptance from Ramanathan and Dickinson [137]. The longwave absorptance formulation includes a Voigt line profile effects for CO$ _2$ and O$ _3$. For the mid-to-upper stratosphere ( $ p\lesssim 10$mb), spectral absorption lines are no longer Lorentzian in shape. To account for the transition to Voigt lines a method described in Kiehl and Briegleb [87] is employed. Essentially the pressure appearing in the mean line width parameter, $ \gamma$,

$\displaystyle \gamma$ $\displaystyle = \gamma_o   \frac{p}{p_0}$ (4.248)

is replaced with

$\displaystyle \gamma$ $\displaystyle = \gamma_0 \left[ \frac{p}{p_0} + \delta \sqrt{\frac{T}{250}} \right] ,$ (4.249)

where $ \delta = 5.0 \times 10^{-3}$ for CO$ _2$ and $ \delta = 2.5
\times 10^{-3}$ for $ O_3$. These values insure agreement with line-by-line cooling rate calculations up to $ p \approx 0.3$ mb.

4.9.2 Water vapor

Water vapor cannot employ the broad-band absorptance method since H$ _2$O absorption extends throughout the entire longwave region. Thus, we cannot factor out the Planck function dependence as in (4.234). The method of Collins et al. [40] is used for water-vapor absorptivities and emissivities. This parameterization replaces the scheme developed by Ramanathan and Downey [138] used in previous versions of the model. The new formulation uses the line-by-line radiative transfer model GENLN3 [57] to generate the absorptivities and emissivities for H$ {}_2$O. In this version of GENLN3, the parameters for H$ _2$O lines have been obtained from the HITRAN2k data base [153], and the continuum is treated with the Clough, Kneizys, and Davies (CKD) model version 2.4.1 [33]. To generate the absorptivity and emissivity, GENLN is used to calculate the transmission through homogeneous atmospheres for H$ _2$O lines alone and for H$ _2$O lines and continuum. The calculation is done for a five dimensional parameter space with coordinates equaling the emission temperature, path temperature, precipitable water, effective relative humidity, and pressure. The limits for each coordinate span the entire range of instantaneous values for the corresponding variable from a 1-year control integration of CAM 3.0. The resulting tables of absorptivity and emissivity are then read into the model for use in the longwave calculations. The overlap treatment between water vapor and other gases is described in Ramanathan and Downey [138].

The absorptivity and emissivity can be split into terms for the window and non-window portions of the infrared spectrum. The window is defined as 800-1200 cm$ {}^{-1}$, and the non-window is the remainder of the spectrum between 20 to 2200 cm$ {}^{-1}$. Outside the mid-infrared window (the so-called non-window region), the H$ _2$O continuum is dominated by the foreign component [34]. The foreign continuum absorption has the same linear scaling with water vapor path as line absorption, and thus in the non-window region the line and continuum absorption are combined in a single expression. In the window region, where the self-broadened component of the continuum is dominant, the line and continuum absorption have different scalings with the amount of water vapor and must be treated separately. The formalism is identical for the absorptivity and emissivity, and for brevity only the absorptivity is discussed in detail. The absorptivity is decomposed into two terms:

$\displaystyle A(p_1,p_2) \simeq A_{w}(p_1,p_2)+ A_{nw}(p_1,p_2)\ ,$ (4.250)

where $ A_{w}(p_1,p_2)$ is the window component and $ A_{nw}(p_1,p_2)$ is the non-window component for the portion of the atmosphere bounded by pressures $ p_1$ and $ p_2$.

Let $ {\widetilde A}_{nw}(i)$ represent the total non-window absorption for a homogeneous atmosphere characterized by a set of scaling parameters $ i$. Scaling theory is a relationship between an inhomogeneous path and an equivalent homogeneous path with nearly identical line absorption for the spectral band under consideration [62]. Scaling theory is used to reduce the parameter space of atmospheric conditions that have to be evaluated. The equivalent pressure, temperature, and absorber amount are calculated using the standard Curtis-Godson scaling theory for absorption lines [61,44]. In addition, we retain explicit dependence on the emission temperature of the radiation following Ramanathan and Downey [138], and we introduce dependence on an equivalent relative humidity. It follows from Curtis-Godson scaling theory that

$\displaystyle A_{nw}(p_1,p_2)\simeq {\widetilde A}_{nw}(l_{nw}) .$ (4.251)

In the following expressions, a tilde denotes a parameter derived using scaling theory for the equivalence between homogeneous and inhomogeneous atmospheres. The subscript $ b$ denotes a parameter which depends upon the spectral band under consideration. The set of scaling parameters that determine the total non-window absorption are labeled:

$\displaystyle l_{nw} = \left[\widetilde {U_{nw}},\widetilde {P_{nw}},T_e,\widetilde {T_p},\widetilde \rho \right] .$ (4.252)

Here $ \widetilde
{U_{nw}}$ is the pressure-weighted precipitable water, $ \widetilde {P_{nw}}$ is the scaled atmospheric pressure, $ T_e$ is the emission temperature of radiation, $ \widetilde {T_p}$ is the absorber weighted path temperature, and $ \widetilde
\rho $ is the scaled relative humidity. The subscript $ (b=)nw$ indicates that the quantities are evaluated for the non-window.

The absorber-weighted path temperature is:

$\displaystyle \widetilde {T_p}= {1 \over W} \> \int_{p_1}^{p_2}T(p)\> dW(p) ,$ (4.253)

where $ T(p)$ is the thermodynamic temperature of the atmosphere at pressure $ p$. The H$ _2$O path or precipitable water is:
$\displaystyle W$ $\displaystyle =$ $\displaystyle \int_{p_1}^{p_2}dW(p) \qquad [g/cm{}^{2}]$ (4.254)
$\displaystyle dW(p)$ $\displaystyle =$ $\displaystyle q(p)\> dp /
g ,$  

where $ q(p)$ is the specific humidity at pressure $ p$ and $ g$ is the acceleration of gravity. The H$ _2$O path and pressure for a homogeneous atmosphere with equivalent line absorption are [62]
$\displaystyle \widetilde {W_{b}}$ $\displaystyle =$ $\displaystyle \int_{p_1}^{p_2}{{\phi_{b}(T)} \over
{\phi_{b}\left(\widetilde {T_p}\right)}}\>dW(p)$ (4.255)
$\displaystyle \widetilde {P_{b}}$ $\displaystyle =$ $\displaystyle {1 \over {\widetilde {W_{b}}}} \int_{p_1}^{p_2}{{\psi_{b}(T)} \over
{\psi_{b}\left(\widetilde {T_p}\right)}}\>p\>dW(p)
,$ (4.256)

$\displaystyle \phi_{b}(T)$ $\displaystyle =$ $\displaystyle \sum_{k=1}^N S_k(T)$ (4.257)
$\displaystyle \psi_{b}(T)$ $\displaystyle =$ $\displaystyle \left\lbrace\sum_{k=1}^N \left[ S_k(T)
\alpha_k(T)\right]^{1/2} \right\rbrace^2
.$ (4.258)

The factor $ S_k(T)$ is the line strength for each line $ k$ in the spectral interval under consideration. The characteristic width of each line at a reference pressure $ p_0$ and specific humidity $ q_0$ is $ \alpha_k(T)$. It is convenient to calculate the absorptance in terms of a pressure-weighted H$ _2$O path

$\displaystyle U = \int_{p_1}^{p_2}{p \over p_0} \> dW(p)$ (4.259)

The equivalent pressure-weighted H$ _2$O path is simply

$\displaystyle \widetilde {U_{b}}= {\widetilde {P_{b}}\over p_0} \> \widetilde {W_{b}}$ (4.260)

Although the relative humidity (or H$ _2$O vapor pressure) is not included in standard Curtis-Godson scaling theory, it must be treated as an independent parameter since the vapor pressure determines the self-broadening of lines and the strength of the self-continuum. The effective relative humidity $ \widetilde
\rho $ is defined in terms of an effective H$ _2$O specific humidity $ \widetilde {q}$ and saturation specific humidity $ \widetilde {q_s}$ along the path:

$\displaystyle \widetilde
\rho$ $\displaystyle =$ $\displaystyle \widetilde {q}\over \widetilde {q_s}$ (4.261)
$\displaystyle \widetilde {q}$ $\displaystyle =$ $\displaystyle {g \> W} \over {p_2 -
p_1}$ (4.262)
$\displaystyle \widetilde {q_s}$ $\displaystyle =$ $\displaystyle {\epsilon\> e_s(\widetilde {T_p})} \over {\widetilde P - (1 -
\epsilon) e_s(\widetilde {T_p})}$ (4.263)
$\displaystyle \widetilde P$ $\displaystyle =$ $\displaystyle {p_0\>U \over W}$ (4.264)

where $ e_s(T)$ is the saturation vapor pressure at temperature $ T$, $ \widetilde P$ is an effective pressure, and $ \epsilon = 0.622$ is the ratio of gas constants for air and water vapor.

The window term $ A_{w}(p_1,p_2)$ requires a special provision for the different path parameters for the lines and continuum. Let

$\displaystyle {\widetilde A}_w(i)$ $\displaystyle =$ $\displaystyle \hbox{absorptivity for path parameters $i$, lines
and continuum}$ (4.265)
$\displaystyle {\widetilde A}'_w(i)$ $\displaystyle =$ $\displaystyle \hbox{absorptivity for path
parameters $i$, lines only}$  

The set of parameters for the line absorption in the window region are:

$\displaystyle l_w = \left[\widetilde {U_{w}},\widetilde {P_{w}},T_e,\widetilde {T_p},\widetilde \rho \right]$ (4.266)

The set of scaling parameters that determine the continuum absorption in the window are:

$\displaystyle c_w = \left[U',\widetilde {P_{w}},T_e,\widetilde {T_p},\widetilde \rho \right]$ (4.267)

For the continuum, the pressure-weighted path length is calculated using:

$\displaystyle U' = {\epsilon \over \widetilde {q}} \> {C_s({\bar\nu},T_{ref}) \over C_s({\bar\nu},\widetilde {T_p})} \> U_c$ (4.268)

where $ T_{ref}= 296K$ is a reference temperature, $ {\bar\nu}$ is a suitably chosen wavenumber inside the window, $ U_c$ is the self-continuum path length, and $ {C_s(\nu,T)}$ is the self continuum absorption coefficient. The self-continuum path length may be approximated by

$\displaystyle U_c = \int_{p_1}^{p_2}{q \over \epsilon} \> {p \over p_0} \> {C_s({\bar\nu},T) \over C_s({\bar\nu},T_{ref})} \> dW(p)$ (4.269)

The lines-only absorptivity can be written in terms of a line transmission factor $ L(i)$ and an asymptotic absorptivity $ A_{w,\infty}$ in the limit of a black-body atmosphere. $ A_{w,\infty}$ is a function only of $ T_e$ [138]. The relationship is

$\displaystyle {\widetilde A}'_w(i) = A_{w,\infty}[1 - L(i)]$ (4.270)

Define an effective continuum transmission $ C(i)$ by setting

$\displaystyle {\widetilde A}_w(i) = A_{w,\infty}[1 - L(i) C(i)]$ (4.271)

We approximate the window absorptivity by:

$\displaystyle A_{w}(p_1,p_2)\simeq A_{w,\infty}[1 - L(l_w) C(c_w)]$ (4.272)

This approximation for $ A_{w}(p_1,p_2)$ can be cast entirely in terms of the absorptivities defined in equation 4.265. From equations 4.270 and 4.271, the line and continuum transmission are:
$\displaystyle L(l_w)$ $\displaystyle =$ $\displaystyle 1 - {{\widetilde A}'_w(l_w) \over A_{w,\infty}}$ (4.273)
$\displaystyle C(c_w)$ $\displaystyle =$ $\displaystyle {{A_{w,\infty}- {\widetilde A}_w(c_w)} \over {A_{w,\infty}- {\widetilde A}'_w(c_w)}}$  

In the presence of stratospheric volcanic aerosols, the expressions for the absorptivity become:

$\displaystyle A_{nw}(p_1,p_2)$ $\displaystyle \simeq$ $\displaystyle A_{nw,\infty}\left[1 - \left(1 - {{\widetilde A}_{nw}(l_{nw}) \over
A_{nw,\infty}} \right) {\cal T}^{nw}_{H_2SO_4} \right]$  
$\displaystyle A_{w}(p_1,p_2)$ $\displaystyle \simeq$ $\displaystyle A_{w,\infty}[1 - L(l_w) C(c_w) {\cal T}^{w}_{H_2SO_4}]$ (4.274)

The volcanic transmission factor is

$\displaystyle {\cal T}^{b}_{H_2SO_4} = {\bar T}^{b}_{H_2SO_4} = \exp\left(-D \kappa_{b,volc} W_{volc}\right)$ (4.275)

where $ D = 1.66$ is the diffusivity factor, $ \kappa_{b,volc}$ is an effective specific extinction for the band, and $ W_{volc}$ is the mass path of the volcanic aerosols. The extinction $ \kappa_{b,volc}$ has been adjusted iteratively to reproduce the heating rates calculated using the spectral bands in the original [138] parameterization. This completes the set of approximations used to calculate the absorptivity (and by extension the emissivity).

4.9.3 Trace gas parameterizations

Methane.    The radiative effects of methane are represented by the last term in (4.241). We re-write this in terms of the absorptivity due to methane as

$\displaystyle \int_{1170}^{1500} { (1-{\cal T}_{CH_4}{\cal T}_{N_2O}^3{\cal
...F(B_\nu)d\nu} = \int {(1-{\cal T}_{H_2O}{\cal T}^{nw}_{H_2SO_4})F(B_\nu)d\nu} +$      
$\displaystyle \int {{\cal A}_{CH_4}{\cal T}_{H_2O}{\cal T}^{5}_{H_2SO_4}F(B_\nu...
A}_{N_2O}^3{\cal T}_{CH_4}{\cal T}_{H_2O}{\cal T}^{5}_{H_2SO_4}F(B_\nu)d\nu}$     (4.276)

Note that this expression also incorporates the absorptance due to the 7.7 micron band of nitrous oxide as well. The first term is due to the rotation band of water vapor and is already accounted for in the CAM 3.0 radiation model by the parameterization described in Ramanathan and Downey [138]. The second term in (4.276) accounts for the absorptance due to the 7.7 micron band of methane. The spectroscopic parameters are from Donner and Ramanathan [51]. In terms of the broad band approximation we have,

$\displaystyle \int {{\cal A}_{CH_4}{\cal T}_{H_2O}{\cal T}^{5}_{H_2SO_4}F(B_\nu)d\nu} \approx A_{CH_4}\bar T_{H_2O}{\bar T}^{5}_{H_2SO_4}F(B_{\bar \nu})$ (4.277)

where according to (4.235),

$\displaystyle A_{CH_4}=6.00444\sqrt {T_p}\ln \left\{ 1+\frac{u} {\sqrt {4+u(1+1/\beta)}} \right\}$ (4.278)

where $ T_p$ is a path weighted temperature,

$\displaystyle T_p = \frac{\int {T(p)dp}}{\int {dp}}$ (4.279)

The dimensionless path length is,

$\displaystyle u = \frac{D \; 8.60957 \times 10^4}{g} \int {\frac{\mu_{CH_4}}{\sqrt T}dp}$ (4.280)

and the mean line width factor is,

$\displaystyle \beta =2.94449 \frac{\int { \frac{1}{T}\left( \frac{P}{P_0} \right) \mu_{CH_4}dp}}{\int {\frac{1}{\sqrt T}\mu_{CH_4}dp}}$ (4.281)

where $ \mu_{CH_4}$ is the mass mixing ratio of methane, $ T$ is the local layer temperature in Kelvin and $ P$ is the pressure in Pascals, and $ P_0$ is $ 1\times 10^5$ Pa. $ D$ is a diffusivity factor of 1.66. The water vapor overlap factor for this spectral region is,

$\displaystyle \bar T_{H_2O}$ $\displaystyle ={\rm exp} (-U_{H_2O})$ (4.282)


$\displaystyle U_{H_2O}$ $\displaystyle = D \int {\mu_{H_2O}\left( \frac{P}{P_0} \right)\frac{dp}{g}}$ (4.283)

and $ \mu_{H_2O}$ is the mass mixing ratio of water vapor.

Nitrous Oxide.    For nitrous oxide there are three absorption bands of interest: 589, 1168 and 1285 cm$ ^{-1}$ bands. The radiative effects of the 1285 cm$ ^{-1}$ band is given by the last term in (4.276),

$\displaystyle \int {{\cal A}_{N_2O}^3{\cal T}_{CH_4}{\cal T}_{H_2O}{\cal T}^{5}...
...approx A_{N_2O}^3\bar T_{CH_4}\bar T_{H_2O}{\bar T}^{5}_{H_2SO_4}F(B_{\bar\nu})$ (4.284)

The absorptance for the 1285 cm$ ^{-1}$ N$ _2$O band is given by

$\displaystyle A_{N_2O}^3=2.35558\sqrt {T_p}\ln \left\{ 1 + \frac{u_0^3}{\sqrt {4+u_0^3(1+1/\beta_0^3)}} + \frac{u_1^3}{\sqrt {4+u_1^3(1+1/\beta_1^3)}} \right\}$ (4.285)

where $ u_0^3$, $ \beta_0^3$ account for the fundamental transition, while $ u_1^3$, $ \beta_1^3$ account for the first ``hot'' band transition. These parameters are defined as

$\displaystyle u_0^3$ $\displaystyle =D \; 1.02346 \times 10^5\int {\frac{\mu_{N_2O}}{\sqrt {T}}\frac{dp}{g}}$ (4.286)


$\displaystyle \beta_0^3$ $\displaystyle = 19.399 \frac{\int {\frac{1}{\sqrt {T}}\left( \frac{P}{P_0} \right)du_0}} {\int {du^3_0}}$ (4.287)

While the ``hot'' band parameters are defined as

$\displaystyle u_1^3$ $\displaystyle = D \; 2.06646\times 10^5 \int {\frac{1}{\sqrt {T}}e^{-847.36/T} \mu_{N_2O}\frac{dp}{g}}$ (4.288)


$\displaystyle \beta_1^3$ $\displaystyle = 19.399 \frac{ \int{ \frac{1}{\sqrt {T}}\left( \frac{P}{P_{0}} \right) du^3_1}}{\int {du^3_1}}$ (4.289)

The overlap factors in (4.284) due to water vapor is the same factor defined by (4.282), while the overlap due to methane is obtained by using the definition of the transmission factor in terms of the equivalent width [136].

$\displaystyle \bar T_{CH_4}=e^{-A_{CH_4}/2A_0}$ (4.290)

Substitution of (4.278) into (4.284) leads to,

$\displaystyle \bar T_{CH_4}=\frac{1}{1+0.02\frac{u}{\sqrt {4+u(1+1/\beta)}}}$ (4.291)

where $ u$ and $ \beta$ are given by (4.280) and (4.281), respectively, and the 0.02 factor is an empirical constant to match the overlap effect obtained from narrow band model benchmark calculations. This factor can physically be justified as accounting for the fact that the entire methane band does not overlap the N$ _2$O band.

The 1168 cm$ ^{-1}$ N$ _2$O band system is represented by the seventh term on the RHS of (4.241). This term can be re-written as

$\displaystyle \int \limits_{1120}^{1170}{ (1-{\cal T}_{CFC12}^4{\cal
...)F(B_\nu)d\nu} = \int {(1-{\cal
T}_{H_2O}{\cal T}^{w}_{H_2SO_4})F(B_\nu)d\nu} +$      
$\displaystyle \int {{\cal A}_{CFC12}^4{\cal
T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_...
T}_{CFC12}^4{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_\nu)d\nu}$     (4.292)

where the last term accounts for the 1168 cm$ ^{-1}$ N$ _2$O band. For the broad band formulation this expression becomes,

$\displaystyle \int {{\cal A}_{N_2O}^2{\cal T}_{CFC12}^4{\cal T}_{H_2O}{\cal T}^...
...x A_{N_2O}^2\bar T_{CFC12}^4\bar T_{H_2O}{\bar T}^{4}_{H_2SO_4}F(B_{\bar \nu })$ (4.293)

The band absorptance for the 1168 cm$ ^{-1}$ N$ _2$O band is given by

$\displaystyle A_{N_2O}^2=2.54034\sqrt {T_p}\ln \left\{ 1+\frac{u_0^2} {\sqrt {4+u_0^2(1+1/{\beta_0^2}}} \right\}$ (4.294)

where the fundamental band path length and mean line parameters can be simply expressed in terms of the parameters defined for the 1285 cm$ ^{-1}$ band (eq. 4.286-4.287).

$\displaystyle u_0^2$ $\displaystyle =0.0333767u_0^3$ (4.295)


$\displaystyle \beta _0^2$ $\displaystyle =0.982143\beta_0^3$ (4.296)

Note that the 1168 cm$ ^{-1}$ band does not include a ``hot'' band transition. The overlap by water vapor includes the effects of water vapor rotation lines, the so called ``e-type'' and ``p-type'' continua (e.g. Roberts et al. [150]). The combined effect of these three absorption features is,

$\displaystyle \bar T_{H_2O}=\bar T_l\bar T_e\bar T_p$ (4.297)

where the contribution by line absorption is modeled by a Malkmus model formulation,

$\displaystyle \bar T_l=\exp \left\{ -\delta_1 \bar \Pi \left( \sqrt {1+\delta_2 \frac{\bar u_l}{\bar \Pi}}-1 \right) \right\}$ (4.298)

where $ \delta_1$ and $ \delta_2$ are coefficients that are obtained by fitting (4.298) to the averaged transmission from a 10 cm$ ^{-1}$ narrow band Malkmus. The path length $ \bar u_l$ is,

$\displaystyle \bar u_l$ $\displaystyle = D \; \bar \Phi \int {\rho_w \frac{dP}{g}}$ (4.299)


$\displaystyle \bar \Pi$ $\displaystyle =\left( \frac{P}{P_0} \right)\left( \frac{\bar\Psi}{\bar\Phi} \right),$ (4.300)

where $ \bar\Phi$ and $ \bar\Psi$ account for the temperature dependence of the spectroscopic parameters [151]
$\displaystyle \bar \Psi$ $\displaystyle =$ $\displaystyle e^{-\alpha \left\vert {T_p-250} \right\vert-\beta \left\vert
{T_p-250} \right\vert^2}$ (4.301)
$\displaystyle \bar \Phi$ $\displaystyle =$ $\displaystyle e^{-\alpha '\left\vert {T_p-250} \right\vert-\beta '\left\vert
{T_p-250} \right\vert^2}$ (4.302)

The coefficients for various spectral intervals are given in Table 4.3. The transmission due to the e-type continuum is given by

$\displaystyle \bar T_e$ $\displaystyle =e^{-\delta_3\bar u_e}$ (4.303)

where the path length is defined as

$\displaystyle \bar u_e$ $\displaystyle =\frac{D}{P_0\varepsilon g} \int {e^{1800(\frac{1}{T}- \frac{1}{296}})w_{H_2O}^2PdP}$ (4.304)

The p-type continuum is represented by

$\displaystyle T_p$ $\displaystyle =e^{-\delta_4\bar u_p}$ (4.305)


$\displaystyle \bar u_p$ $\displaystyle = \frac{D}{gP_0} \int {e^{1800(\frac{1}{T}-\frac{1}{296})} w_{H_2O}PdP}$ (4.306)

The factors $ \delta_1$, $ \delta_2$, $ \delta_3$ and $ \delta_4$ are listed for specific spectral intervals in Table 4.4.

Table 4.3: Coefficients for the Temperature Dependence Factors in (4.301) and (4.302).
Index $ \nu_1-\nu_2$ $ \alpha$ $ \beta$ $ \alpha'$ $ \beta'$
1 750 - 820 2.9129e-2 -1.3139e-4 3.0857e-2 -1.3512e-4
2 820 - 880 2.4101e-2 -5.5688e-5 2.3524e-2 -6.8320e-5
3 880 - 900 1.9821e-2 -4.6380e-5 1.7310e-2 -3.2609e-5
4 900 - 1000 2.6904e-2 -8.0362e-5 2.6661e-2 -1.0228e-5
5 1000 - 1120 2.9458e-2 -1.0115e-4 2.8074e-2 -9.5743e-5
6 1120 - 1170 1.9892e-2 -8.8061e-5 2.2915e-2 -1.0304e-4

Table 4.4: Coefficients for the broad-band water vapor overlap transmission factors.
Index $ \nu_1-\nu_2$ $ \delta_1$ $ \delta_2$ $ \delta_3$ $ \delta_4$
1 750 - 820 0.0468556 14.4832 26.1891 0.0261782
2 820 - 880 0.0397454 4.30242 18.4476 0.0369516
3 880 - 900 0.0407664 5.23523 15.3633 0.0307266
4 900 - 1000 0.0304380 3.25342 12.1927 0.0243854
5 1000 - 1120 0.0540398 0.698935 9.14992 0.0182932
6 1120 - 1170 0.0321962 16.5599 8.07092 0.0161418

The final N$ _2$O band centered at 589 cm$ ^{-1}$ is represented by the first term on the RHS of (4.241),

$\displaystyle {\int\limits_{500}^{750} {(1 - {\cal
T}_{CO_2}^1{\cal T}_{N_2O}^1{\cal T}_{H_2O}{\cal T}^{1}_{H_2SO_4}) F(B_\nu) d\nu} =}$
    $\displaystyle \int {(1-{\cal T}_{CO_2}^1{\cal_T}_{H_2O}{\cal T}^{1}_{H_2SO_4})F...
... A}_{N_2O}^1{\cal T}_{CO_2}^1{\cal T}_{H_2O}{\cal T}^{1}_{H_2SO_4}F(B_\nu)d\nu}$ (4.307)

where the last term in (4.307) represents the radiative effects of the 589 cm$ ^{-1}$ N$ _2$O band,

$\displaystyle \int {{\cal A}_{N_2O}^1{\cal T}_{CO_2}^1{\cal T}_{H_2O}{\cal T}^{...
...ox A_{N_2O}^1\bar T_{CO_2}^1\bar T_{H_2O}{\bar T}^{1}_{H_2SO_4}F(B_{\bar \nu })$ (4.308)

The absorptance for this band includes both the fundamental and hot band transitions,

$\displaystyle A_{N_2O}^1 = 2.65581 \sqrt{T_p} \ln \left\{ 1 + \frac{u_0^1}{\sqr...
...0^1(1 + 1/\beta_0^1)}} + \frac{u_1^1}{\sqrt{4+u_1^1(1 + 1/\beta_1^1)}} \right\}$ (4.309)

where the path lengths for this band can also be defined in terms of the 1285 cm$ ^{-1}$ band path length and mean lines parameters (4.286 - 4.289),

$\displaystyle u_0^1$ $\displaystyle = 0.100090u_0^3$ (4.310)


$\displaystyle \beta_0^1$ $\displaystyle = 0.964282\beta_0^3$ (4.311)


$\displaystyle u_1^1$ $\displaystyle = 0.0992746u_1^3$ (4.312)


$\displaystyle \beta _1^1$ $\displaystyle = 0.964282\beta_1^3$ (4.313)

The overlap effect of water vapor is given by the transmission factor for the 500 to 800 cm$ ^{-1}$ spectral region defined by Ramanathan and Downey [138] in their Table A2. This expression is thus consistent with the transmission factor for this spectral region employed for the water vapor formulation of the first term on the right hand side of (4.307). The overlap factor due to the CO$ _2$ bands near 589 cm$ ^{-1}$ is obtained from the formulation in Kiehl and Briegleb [87],

$\displaystyle \bar T_{CO_2}^1 = \frac{1}{1 + 0.2\frac{u_{CO_2}} {\sqrt{4+u_{CO_2}(1+1/\beta_{CO_2})}}}$ (4.314)

where the functional form is obtained in the same manner as the transmission factor for CH$ _4$ was determined in (4.290). The 0.2 factor is empirically determined by comparing (4.314) with results from 5 cm$ ^{-1}$ Malkmus narrow band calculations. The path length parameters are given by

$\displaystyle u_{CO_2}$ $\displaystyle =\frac{D \; 4.9411\times 10^4(1-e^{-960/T})^3} {\sqrt{T_p}} e^{-960/T} \int {w_{CO_2} \frac{dP}{g}}$ (4.315)


$\displaystyle \beta_{CO_2}$ $\displaystyle = \frac{5.3228}{\sqrt{T_p}} \left\{ \frac{P}{P_0} +5\times e^{-3}\sqrt{\frac{T}{250}\frac{T}{300}} \right\}$ (4.316)

CFCs.    The effects of both CFC11 and CFC12 are included by using the approach of Ramanathan et al. [139]. Thus, the band absorptance of the CFCs is given by

$\displaystyle A_{CFC}=\Delta \nu \left( 1-e^{-D\frac{S}{\Delta \nu}u_{CFC}} \right)$ (4.317)

where $ \Delta\nu$ is the width of the CFC absorption band, $ S$ is the band strength, $ u_{CFC}$ is the abundance of CFC (g cm$ ^{-2}$),

$\displaystyle u_{CFC}=\int {\mu_{CFC}\frac{dp}{g}}$ (4.318)

where $ \mu_{CFC}$ is the mass mixing ratio of either CFC11 or CFC12. $ D$ is the diffusivity factor. In the linear limit $ D=2$, since (4.317) deviates slightly from the pure linear limit we let $ D=1.8$. We account for the radiative effects of four bands due to CFC11 and four bands due to CFC12. The band parameters used in (4.317) for these eighth bands are given in Table 4.5.

The contribution by these CFC absorption bands is accounted for by the following terms in (4.241).

$\displaystyle \int\limits_{750}^{820} {(1-{\cal T}_{CFC11}^1{\cal T}_{H_2O}{\cal T}^{*}_{H_2SO_4})
F(B_\nu)d\nu}$ $\displaystyle =$ $\displaystyle \int {(1-{\cal T}_{H_2O}{\cal T}^{nw}_{H_2SO_4})F(B_\nu)d\nu}$  
  $\displaystyle +$ $\displaystyle \int {{\cal A}_{CFC11}^1{\cal T}_{H_2O}{\cal T}^{*}_{H_2SO_4}F(B_\nu)d\nu}$ (4.319)
$\displaystyle \int\limits_{820}^{880} {(1-{\cal T}_{CFC11}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4})
F(B_\nu)d\nu}$ $\displaystyle =$ $\displaystyle \int {(1-{\cal T}_{H_2O}{\cal T}^{w}_{H_2SO_4})F(B_\nu)d\nu}$  
  $\displaystyle +$ $\displaystyle \int {{\cal A}_{CFC11}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu)d\nu}$ (4.320)
$\displaystyle \int\limits_{880}^{900} {(1-{\cal T}_{CFC12}^1{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4})
F(B_\nu)d\nu}$ $\displaystyle =$ $\displaystyle \int {(1-{\cal T}_{H_2O}{\cal T}^{w}_{H_2SO_4})F(B_\nu)d\nu}$  
  $\displaystyle +$ $\displaystyle \int {{\cal A}_{CFC12}^1{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu)d\nu}$ (4.321)
$\displaystyle \int\limits_{900}^{1000} {(1 -{\cal T}_{CO_2}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\cal
T}_{CFC11}^3{\cal T}_{CFC12}^2) F(B_\nu)d\nu}$ $\displaystyle =$ $\displaystyle \int {(1-{\cal T}_{H_2O}{\cal T}^{w}_{H_2SO_4})F(B_\nu)d\nu}$  
$\displaystyle + \int {{\cal A}_{CFC12}^2{\cal
T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu)d\nu}$ $\displaystyle +$ $\displaystyle \int {{\cal A}_{CFC11}^3{\cal
T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\cal T}_{CFC12}^2F(B_\nu)d\nu}$ (4.322)
  $\displaystyle +$ $\displaystyle \int {{\cal
A}_{CO_2}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\cal T}_{CFC11}^3{\cal T}_{CFC12}^2
$\displaystyle \int\limits_{1000}^{1120} {(1 -{\cal T}_{CO_2}^3{\cal T}_{O_3}{\c...
...{H_2O}{\cal T}^{4}_{H_2SO_4} {\cal T}_{CFC11}^4{\cal T}_{CFC12}^3)F(B_\nu)d\nu}$ $\displaystyle =$ $\displaystyle \int {(1-{\cal T}_{H_2O}{\cal T}^{w}_{H_2SO_4})F(B_\nu)d\nu}$  
$\displaystyle + \int {{\cal
A}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_\nu)d\nu}$ $\displaystyle +$ $\displaystyle \int {{\cal A}_{CO_2}^3{\cal
T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}{\cal T}_{CFC11}^4{\cal
$\displaystyle + \int {{\cal A}_{CFC11}^4{\cal
T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_\nu)d\nu}$ $\displaystyle +$ $\displaystyle \int {{\cal A}_{CFC12}^3{\cal
T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_\nu)d\nu}$ (4.323)

Table 4.5: Band Parameters for the CFCs transmission factors.
Band Number Band Center $ \Delta\nu$ $ S/\Delta\nu$
  (cm$ ^{-1}$) (cm$ ^{-1}$) (cm$ ^2$ gm$ ^{-1}$)
1$ ^1$ 798 50 54.09
2$ ^2$ 846 60 5130.03
3$ ^1$ 933 60 175.005
4$ ^2$ 1085 100 1202.18
1$ ^1$ 889 45 1272.35
2$ ^2$ 923 50 5786.73
3$ ^2$ 1102 80 2873.51
4$ ^2$ 1161 70 2085.59
$ ^1$ Data are from Kagann et al. [82].
$ ^2$ Data are from Varanasi and Chudamani [177].

For the 798 cm$ ^{-1}$ CFC11 band, the absorption effect is given by the second term on the right hand side of (4.319),

$\displaystyle \int {{\cal A}_{CFC11}^1{\cal T}_{H_2O}{\cal T}^{*}_{H_2SO_4}F(B_\nu)d\nu} \approx A_{CFC11}^1\bar T_{H_2O}{\bar T}^{*}_{H_2SO_4}F(B_{\bar \nu })$ (4.324)

where the band absorptance for the CFC is given by (4.317) and the overlap factor due to water vapor is given by (4.297) using the index 1 factors from Tables 4.3 and 4.4. Similarly, the $ 846 \rm {cm}^{-1}$ CFC11 band is represented by the second term on the RHS of (4.320),

$\displaystyle \int {{\cal A}_{CFC11}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu)d\nu} \approx A_{CFC11}^2\bar T_{H_2O}{\bar T}^{3}_{H_2SO_4}F(B_{\bar\nu})$ (4.325)

where the H$ _2$O overlap factor is given by index 2 in Tables 4.3 and 4.4. The 933 cm$ ^{-1}$ CFC11 band is given by the third term on the RHS of (4.322),

$\displaystyle \int {{\cal A}_{CFC11}^3{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\ca...
...approx A_{CFC11}^3\bar T_{H_2O}{\bar T}^{3}_{H_2SO_4}T_{CFC12}^2F(B_{\bar \nu})$ (4.326)

where the H$ _2$O overlap factor is defined as index 4 in Tables 4.3 and 4.4, and the CFC12 transmission factor is obtained from (4.317). The final CFC11 band centered at 1085 cm$ ^{-1}$ is represented by the fourth term on the RHS of (4.323),

$\displaystyle \int{ {\cal A}_{CFC11}^4{\cal T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}...
...pprox A_{CFC11}^4\bar T_{O_3}\bar T_{H_2O}{\bar T}^{4}_{H_2SO_4}F(B_{\bar \nu})$ (4.327)

where the transmission due to the 9.6 micron ozone band is defined similar to (4.314) for CO$ _2$ as

$\displaystyle \bar T_{O_3}= \frac{1}{1+\sum\limits_{i=1}^2 \frac{u_{O_3}^i} {\sqrt{4+u_{O_3}^i(1 + 1/\beta_{O_3}^i)}}}$ (4.328)

where the path lengths are defined in Ramanathan and Dickinson [137]. The H$ _2$O overlap factor is defined by index 5 in Tables 4.3 and 4.4.

For the 889 cm$ ^{-1}$ CFC12 band the absorption is defined by the second term in (4.321) as

$\displaystyle \int { {\cal A}_{CFC12}^1{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu)d\nu} \approx A_{CFC12}^1\bar T_{H_2O}{\bar T}^{3}_{H_2SO_4}F(B_{\bar \nu })$ (4.329)

where the H$ _2$O overlap factor is defined by index 3 of Tables 4.3 and 4.4, and the CFC absorptance is given by (4.317). The 923 cm$ ^{-1}$ CFC12 band is described by the second term in (4.322),

$\displaystyle \int {{\cal A}_{CFC12}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}F(B_\nu) d\nu} \approx A_{CFC12}^2\bar T_{H_2O}{\bar T}^{3}_{H_2SO_4}F(B_{\bar \nu})$ (4.330)

where the H$ _2$O overlap is defined as index 4 in Tables 4.3 and 4.4. The 1102 cm$ ^{-1}$ CFC12 band is represented by the last term on the RHS of (4.323),

$\displaystyle \int{ {\cal A}_{CFC12}^3{\cal T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}...
...pprox A_{CFC12}^3\bar T_{O_3}\bar T_{H_2O}{\bar T}^{4}_{H_2SO_4}F(B_{\bar \nu})$ (4.331)

where the transmission by ozone is described by (4.328) and the H$ _2$O overlap factor is represented by index 5 in Tables 4.3 and 4.4. The final CFC12 band at 1161 cm$ ^{-1}$ is represented by the second term on the RHS of (4.292),

$\displaystyle \int{{\cal A}_{CFC12}^4{\cal T}_{H_2O}{\cal T}^{4}_{H_2SO_4}F(B_\nu) d\nu} \approx A_{CFC12}^4\bar T_{H_2O}{\bar T}^{4}_{H_2SO_4}F(B_{\bar \nu})$ (4.332)

where the H$ _2$O overlap factor is defined as index 6 in Tables 4.3 and 4.4.

Minor CO$ _2$ Bands.    There are two minor bands of carbon dioxide that were added to the CCM3 longwave model. These bands play a minor role in the present day radiative budget, but are very important for high levels of CO$ _2$, such as during the Archean. The first band we consider is centered at 961 cm$ ^{-1}$. The radiative contribution of this band is represented by the last term in (4.322),

$\displaystyle \int {{\cal A}_{CO_2}^2{\cal T}_{H_2O}{\cal T}^{3}_{H_2SO_4}{\cal...
...r T_{H_2O}{\bar T}^{3}_{H_2SO_4}\bar T_{CFC11}^3\bar T_{CFC12}^2F(B_{\bar \nu})$ (4.333)

where the transmission factors for water vapor, CFC11 and CFC12 are defined in the previous section for the 900 to 1000 cm$ ^{-1}$ spectral interval. The absorptance due to CO$ _2$ is given by

$\displaystyle A_{CO_2}^2 = 3.8443\sqrt{T_p}\ln \left\{ 1+\sum\limits_{i=1}^3 \frac{u_i}{\sqrt {4+u_i(1+1/\beta_i)}} \right\}$ (4.334)

where the path length parameters are defined as
$\displaystyle u_1$ $\displaystyle =$ $\displaystyle 3.88984 \times 10^3\alpha (T_{p}) we^{-1997.6/T}$ (4.335)
$\displaystyle u_1$ $\displaystyle =$ $\displaystyle 3.88984 \times 10^3\alpha (T_{p}) we^{-1997.6/T}$ (4.336)
$\displaystyle u_3$ $\displaystyle =$ $\displaystyle 6.50642\times 10^3\alpha(T_{p}) we^{-2989.7/T}$ (4.337)

and the pressure parameter is,
$\displaystyle \beta _1$ $\displaystyle =$ $\displaystyle 2.97558\left( \frac{P}{P_{0}} \right)\frac{1}{\sqrt{T}}$ (4.338)
$\displaystyle \beta_2$ $\displaystyle =$ $\displaystyle \beta _1$ (4.339)
$\displaystyle \beta_3$ $\displaystyle =$ $\displaystyle 2\beta_1$ (4.340)


$\displaystyle \alpha (T_p)= \frac{\left( 1-e^{-1360.0/T_p} \right)^3}{\sqrt{T_p}}$ (4.341)

The CO$ _2$ band centered at 1064 cm$ ^{-1}$ is represented by the third term on the RHS of (4.323),

$\displaystyle \int {{\cal A}_{CO_2}^3{\cal T}_{O_3}{\cal T}_{H_2O}{\cal T}^{4}_...
... T_{H_2O}{\bar T}^{4}_{H_2SO_4} \bar T_{CFC11}^4\bar T_{CFC12}^3F(B_{\bar \nu})$ (4.342)

where the transmission factors due to ozone, water vapor, CFC11 and CFC12 are defined in the previous section. The absorptance due to the 1064 cm$ ^{-1}$ CO$ _2$ band is given by

$\displaystyle A_{CO_2}^3 = 3.8443\sqrt{T_p}\ln \left\{ 1+\sum\limits_{i=1}^3 \frac{u_i}{\sqrt {4+u_i(1+1/\beta_i)}} \right\}$ (4.343)

where the dimensionless path length is defined as

$\displaystyle u_1$ $\displaystyle = 3.42217\times 10^3\alpha (T_{p}) we^{-1849.7/T}$ (4.344)
$\displaystyle u_2$ $\displaystyle = 6.02454\times 10^3\alpha (T_{p}) we^{-2782.1/T}$ (4.345)
$\displaystyle u_3$ $\displaystyle = 5.53143\times 10^3\alpha (T_{p}) we^{-3723.2/T}$ (4.346)


$\displaystyle \alpha (T_p)$ $\displaystyle = \frac{\left( 1-e^{-1540.0/T_p} \right)^3}{\sqrt{T_p}}$ (4.347)

The pressure factor, $ \beta_1$, for (4.343) is the same as defined in (4.338), while the other factors are,

$\displaystyle \beta_2$ $\displaystyle =$ $\displaystyle 2\beta_1$ (4.348)
$\displaystyle \beta_3$ $\displaystyle =$ $\displaystyle \beta_2$ (4.349)

In the above expressions, $ w$ is the column mass abundance of CO$ _2$,

$\displaystyle w=\int {\mu _{CO_2}\frac{dP}{g}} = \frac{\mu _{CO_2}}{g}\Delta P$ (4.350)

where $ \mu_{CO_2}$ is the mass mixing ratio of CO$ _2$ (assumed constant).

4.9.4 Mixing ratio of trace gases

The mixing ratios of methane, nitrous oxide, CFC11 and CFC12 are specified as zonally averaged quantities. The stratospheric mixing ratios of these various gases do vary with latitude. This is to mimic the effects of stratospheric circulation on these tracers. The exact latitude dependence of the mixing ratio scale height was based on information from a two dimensional chemical model (S. Solomon, personal communication). In the troposphere the gases are assumed to be well mixed,

$\displaystyle \mu_{CH_4}^0$ $\displaystyle =$ $\displaystyle 0.55241w_{CH_4}$ (4.351)
$\displaystyle \mu_{N_2O}^0$ $\displaystyle =$ $\displaystyle 1.51913w_{N_2O}$ (4.352)
$\displaystyle \mu_{CFC11}^0$ $\displaystyle =$ $\displaystyle 4.69548w_{CFC11}$ (4.353)
$\displaystyle \mu_{CFC12}^0$ $\displaystyle =$ $\displaystyle 4.14307w_{CFC12}$ (4.354)

where $ w$ denotes the volume mixing ratio of these gases. The CAM 3.0 employs volume mixing ratios for the year 1992 based on IPCC [79], $ w_{CH_4} = 1.714 ppmv$, $ w_{N_2O}=0.311 ppmv$, $ w_{CFC11} = 0.280 ppbv$ and $ w_{CFC12}=0.503 ppbv$. The pressure level (mb) of the tropopause is defined as

$\displaystyle p_{trop}=250.0-150.0\cos ^2\phi$ (4.355)

For $ p\leq p_{trop}$, the stratospheric mixing ratios are defined as
$\displaystyle \mu _{CH_4}$ $\displaystyle =$ $\displaystyle \mu _{CH_4}^0\left( \frac{p}{p_{trop}}
\right)^{X_{CH_4}}$ (4.356)
$\displaystyle \mu _{N_2O}$ $\displaystyle =$ $\displaystyle \mu _{N_2O}^0\left( \frac{p}{p_{trop}}
\right)^{X_{N_2O}}$ (4.357)
$\displaystyle \mu _{CFC11}$ $\displaystyle =$ $\displaystyle \mu _{CFC11}^0\left( \frac{p}{p_{trop}}
\right)^{X_{CFC11}}$ (4.358)
$\displaystyle \mu _{CFC12}$ $\displaystyle =$ $\displaystyle \mu _{CFC12}^0\left( \frac{p}{p_{trop}}
\right)^{X_{CFC12}}$ (4.359)

where the mixing ratio scale heights are defined as

$\displaystyle \left. \begin{matrix}X_{CH_4} & = & 0.2353 \ X_{N_2O} & = & 0.34...
...vert \phi \right\vert \ \end{matrix} \right\}\left\vert \phi \right\vert\le 45$ (4.360)


$\displaystyle \left. \begin{matrix}X_{CH_4} & = & 0.2353+0.22549\left\vert \phi...
...vert \phi \right\vert \ \end{matrix} \right\}\left\vert \phi \right\vert\ge 45$ (4.361)

where $ \phi $ is latitude in degrees.

4.9.5 Cloud emissivity

The clouds in CAM 3.0 are gray bodies with emissivities that depend on cloud phase, condensed water path, and the effective radius of ice particles. The cloud emissivity is defined as

$\displaystyle \epsilon_{cld} =1-e^{-D\kappa _{abs}CWP}$ (4.362)

where $ D$ is a diffusivity factor set to 1.66, $ \kappa_{abs}$ is the longwave absorption coefficient ( $ m^2 g^{-1}$), and CWP is the cloud water path ($ g m^{-2}$). The absorption coefficient is defined as

$\displaystyle \kappa _{abs}=\kappa _l\left( {1-f_{ice}} \right)+\kappa _if_{ice}$ (4.363)

where $ \kappa_l$ is the longwave absorption coefficient for liquid cloud water and has a value of 0.090361, such that $ D\kappa_l$ is 0.15. $ \kappa_i$ is the absorption coefficient for ice clouds and is based on a broad band fit to the emissivity given by Ebert and Curry's formulation,

$\displaystyle \kappa_i=0.005 + \frac{1}{r_{ei}}.$ (4.364)

4.9.6 Numerical algorithms and cloud overlap

The treatment of cloud overlap follows Collins [38]. The new parameterizations can treat random, maximum, or an arbitrary combination of maximum and random overlap between clouds. This scheme replaces the treatment in CCM3, which was an exact treatment for random overlap of plane-parallel infinitely-thin gray-body clouds. The new method is an exact treatment for arbitrary overlap among the same type of clouds. It is therefore more accurate than the original matrix method of Manabe and Strickler [123] and improved variants of it [135,107].

If longwave scattering is omitted, the upwelling and downwelling longwave fluxes are solutions to uncoupled ordinary differential equations [62]. The emission from clouds is calculated using the Stefan-Boltzmann law applied to the temperatures at the cloud boundaries. The cloud boundaries correspond to the interfaces of the model layers. This approximation greatly simplifies the mathematical form of the flux solutions since the clouds can be treated as boundary conditions for the differential equations. The approximation becomes more accurate as the clouds become more optically thick.

The solutions are formulated in terms of the same conversion of vertical cloud distributions to binary cloud profiles used for the shortwave calculations (p. [*]). First consider the flux boundary conditions for a maximum-overlap region $ j$. The downward flux at the upper boundary of the region is spatially heterogeneous and has terms contributed by all the binary configurations above the region. Similarly, the upward flux at the lower boundary of the region has terms contributed by all the binary configurations below the region. The fluxes within the region are area-weighted sums of the fluxes calculated for all possible combinations of these boundary terms and the cloud configurations within the region. Fortunately the arithmetic can be simplified because the solutions to the longwave equations are linear in the boundary conditions. Therefore the downward (upward) fluxes can be computed by summing the solutions for each configuration in the region for a single boundary condition given by the area-averaged fluxes at the region interfaces denoted by $ \bar{F}^{\downarrow}(i_{{j},\min})$ ( $ \bar{F}^{\uparrow}(i_{{j},\max})$). The mathematics is explained in Collins [38]. In the absorptivity-emissivity method, the boundary conditions are included in the solution using the emissivity array. In the standard formulation [138,122] used in CAM 3.0, this array is only defined for boundary conditions at the top of the model domain for computational economy. It is not possible to treat arbitrary flux boundary conditions inside the domain (e.g., $ \bar{F}^{\downarrow}(i_{{j},\min})$) using the emissivity array. However, the flux boundary conditions $ \bar{F}^{\downarrow}(i_{{j},\min})$ and $ \bar{F}^{\uparrow}(i_{{j},\max})$ are mathematically equivalent to the fluxes from a single ``pseudo'' cloud deck above and below the region, respectively. The pseudo clouds have unit area and occupy a single model layer. The vertical positions and emissivities of these clouds are chosen so that the net area-mean fluxes incident on the top and bottom of the region equal $ \bar{F}^{\downarrow}(i_{{j},\min})$ and $ \bar{F}^{\uparrow}(i_{{j},\max})$. With the introduction of the pseudo clouds, the fluxes inside each maximum-overlap region can be calculated using the standard absorptivity-emissivity formulation.

The total upward and downward mean fluxes at a layer $ i$ within a maximum-overlap region $ j$ are given by:

$\displaystyle \bar{F}^{\uparrow}(i)$ $\displaystyle =$ $\displaystyle \sum_{k_{j}= 1}^{n_{j}+1} \tilde{A}_{{j},k_{j}}
$\displaystyle \bar{F}^{\downarrow}(i)$ $\displaystyle =$ $\displaystyle \sum_{k_{j}= 1}^{n_{j}+1} \tilde{A}_{{j},k_{j}}
\bar{F}[k_{j}]^{\downarrow}(i)$ (4.365)

where $ \bar{F}[k_{j}]^{\uparrow}(i)$ and $ \bar{F}[k_{j}]^{\downarrow}(i)$ are the upward and downwelling fluxes for the cloud configuration $ \tilde{\cfrac }_{{j},k_{j}}$. The symbols required to write these fluxes are defined in Table 4.6.

Table 4.6: Definition of terms in fluxes.
$ \sigma$ Stefan-Boltzmann constant
$ p$ pressure
$ p_t(i)$ pressure at top of layer $ i$
$ p_b(i)$ pressure at bottom of layer $ i$ ( $ p_b(i) > p_t(i)$)
$ T(p)$ temperature at pressure $ p$
$ B(p)$ $ \sigma T^4(p)$
$ i_{p,j}^{\downarrow}$ layer containing pseudo cloud for $ \bar{F}^{\downarrow}(i_{{j},\min})$ b.c.
$ i_{p,j}^{\uparrow}$ layer containing pseudo cloud for $ \bar{F}^{\uparrow}(i_{{j},\max})$ b.c.
$ \epsilon_{cld}(i)$ emissivity of cloud in layer $ i$
$ \epsilon_{p,j}(i)$ emissivity of pseudo clouds at $ i = i_{p,j}^{\downarrow}$ and $ i_{p,j}^{\uparrow}$
$ \alpha(p,p')$ clear-sky absorptivity from pressure $ p'$ to $ p$
$ F_{clr}^\downarrow(i)$ downwelling clear-sky flux at layer $ i$
$ F_{clr}^\uparrow(i)$ upwelling clear-sky flux at layer $ i$
$ t_{j,k_{j}}^{\uparrow\downarrow}(i)$ weights for up/downwelling clear-sky flux at layer $ i$
$ T_{j,k_{j}}^{\uparrow\downarrow}(i,i')$ weights for up/downwelling flux at layer $ i$ from cloud at $ i'$

The downward and upward fluxes for each configuration can be derived by iterating the longwave equations from TOA and the surface to the layer $ i$. At each iteration, the solutions are advanced between successive cloud layers. The final form of the fluxes in configuration $ \tilde{\cfrac }_{{j},k_{j}}$ is:

$\displaystyle \bar{F}[k_{j}]^{\uparrow}(i)$ $\displaystyle =$ $\displaystyle F_{clr}^\uparrow(i)t_{j,k_{j}}^{\uparrow}(i) +$ (4.366)
    $\displaystyle \sum_{i' = i}^N
\left\{ B\left(p_t(i')\right) -
{dB(p') \over dp'} dp' \right\}
$\displaystyle \bar{F}[k_{j}]^{\downarrow}(i)$ $\displaystyle =$ $\displaystyle F_{clr}^\downarrow(i)t_{j,k_{j}}^{\downarrow}(i) +$ (4.367)
    $\displaystyle \sum_{i' = 1}^i
\left\{ B\left(p_b(i')\right) +
{dB(p') \over dp'} dp' \right\}

The clear-sky and cloudy-sky weights are:
$\displaystyle t_{j,k_{j}}^{\uparrow}(i)$ $\displaystyle =$ $\displaystyle \prod_{l = i}^N \left[1-\tilde{\epsilon}_{{j},k_{j}}(l)\right]$ (4.368)
$\displaystyle t_{j,k_{j}}^{\downarrow}(i)$ $\displaystyle =$ $\displaystyle \prod_{l = 1}^i \left[1-\tilde{\epsilon}_{{j},k_{j}}(l)\right]$ (4.369)
$\displaystyle T_{j,k_{j}}^{\uparrow}(i,i')$ $\displaystyle =$ $\displaystyle \tilde{\epsilon}_{{j},k_{j}}(i')\prod_{l = i}^{i'-1}
\left[1-\tilde{\epsilon}_{{j},k_{j}}(l)\right]$ (4.370)
$\displaystyle T_{j,k_{j}}^{\downarrow}(i,i')$ $\displaystyle =$ $\displaystyle \tilde{\epsilon}_{{j},k_{j}}(i')\prod_{l = i'+1}^{i} \left[1-\tilde{\epsilon}_{{j},k_{j}}(l)\right]$ (4.371)
$\displaystyle \tilde{\epsilon}_{{j},k_{j}}(l)$ $\displaystyle =$ $\displaystyle \left\{ \begin{array}{ll}
\epsilon_{cld}(l)\tilde{\cfrac }_{{j},k...
... \mbox{if $l = i_{p,j}^{\uparrow}$} \ 0
& \mbox{otherwise}
\end{array} \right.$ (4.372)

The longwave atmospheric heating rate is obtained from

$\displaystyle Q_{\ell w} (p_k) = \frac{g}{c_p} \frac{\bar{F}^{\uparrow}(k+1) - ...
...arrow}(k+1) - \bar{F}^{\uparrow}(k) + \bar{F}^{\downarrow}(k)}{p_{k+1} - p_k} .$ (4.373)

which is added to the nonlinear term $ (Q)$ in the thermodynamic equation.

The full calculation of longwave radiation (which includes heating rates as well as boundary fluxes) is computationally expensive. Therefore, modifications to the longwave scheme were developed to improve its efficiency for the diurnal framework. For illustration, consider the clear-sky fluxes defined in (4.229) and (4.230). Well over 90% of the longwave computational cost involves evaluating the absorptivity $ \alpha$ and emissivity $ \epsilon$. To reduce this computational burden, $ \alpha$ and $ \epsilon$ are computed at a user defined frequency that is set to every 12 model hours in the standard configuration, while longwave heating rates are computed at the diurnal cycle frequency of once every model hour.

Calculation of $ \alpha$ and $ \epsilon$ with a period longer than the evaluation of the longwave heating rates neglects the dependence of these quantities on variations in temperature, water vapor, and ozone. However, variations in radiative fluxes due to changes in cloud amount are fully accounted for at each radiation calculation, which is regarded to be the dominant effect on diurnal time scales. The dominant effect on the heating rates of changes in temperature occurs through the Planck function and is accounted for with this method.

The continuous equations for the longwave calculations require a sophisticated vertical finite-differencing scheme due to the integral term $ \int \alpha dB$ in Equations (4.229)-(4.230). The reason for the additional care in evaluating this integral arises from the nonlinear behavior of $ \alpha$ across a given model layer. For example, if the flux at interface $ p_k$ is required, an integral of the form $ \int^{p_k}_{p_s} \alpha (p', p_k) dB(p')$ must be evaluated. For the nearest layer to level $ p_k$, the following terms will arise:

$\displaystyle \int^{p_k}_{p_{k+1}} {\alpha (p', p_k) dB (p')} = \frac{\left [ \...
...k+1}, p_k) + \alpha(p_k, p_k) \right]}{2} \left [B(p_k) - B(p_{k+1}) \right ] ,$ (4.374)

employing the trapezoidal rule. The problem arises with the second absorptivity $ \alpha (p_k, p_k)$, since this term is zero. It is also known that $ \alpha$ is nearly exponential in form within a layer. Thus, to accurately account for the variation of $ \alpha(p,p')$ across a layer, many more grid points are required than are available in CAM 3.0. The nearest layer must, therefore, be subdivided and $ \alpha$ must be evaluated across the subdivided layers. The algorithm that is employed in is to use a trapezoid method for all layers except the nearest layer. For the nearest layer a subdivision, as illustrated in Figure 4.3, is employed.
Figure 4.3: Subdivision of model layers for radiation flux calculation

For the upward flux, the nearest layer contribution to the integral is evaluated from

$\displaystyle \int^{p^{k+1}_H}_{p^k_H} \alpha dB(p') = \alpha_{22} \left[B(p^{k+1}_H) - B(p^k) \right] + \alpha_{21} \left[B(p^k) - B(p^k_H) \right] ,$ (4.375)

while for the downward flux, the integral is evaluated according to

$\displaystyle \int^{p^k_H}_{p^{k+1}_H} \alpha dB(p') = \alpha_{11} \left[B (p^k) - B(p^k_H) \right] + \alpha_{12} \left[B(p^{k+1}_H) - B(p^k) \right] .$ (4.376)

The $ \alpha_{ij},\; i = 1, 2;\; j = 1, 2,$ are absorptivities evaluated for the subdivided paths shown in Figure 4.3. The path-length dependence for the absorptivities arises from the dependence on the absorptance $ A(p,p')$ [e.g., Eq. (4.373)]. Temperatures are known at model levels. Temperatures at layer interfaces are determined through linear interpolation in $ \log p$ between layer midpoint temperatures. Thus, $ B(p_k) = \sigma_B T^4_k$ can be evaluated at all required levels. The most involved calculation arises from the evaluation of the fraction of layers shown in Figure 4.3. In general, the absorptance of a layer can require the evaluation of the following path lengths:

$\displaystyle \xi (p_k, p_{k+1})$ $\displaystyle = f(\overline {T}) \overline {p} \Delta p ,$ (4.377)


$\displaystyle u(p_k, p_{k+1})$ $\displaystyle = g (\overline {T}) \Delta p,$ (4.378)


$\displaystyle \beta (p_k, p_{k+1})$ $\displaystyle = h (\overline {T}) \overline {p} ,$ (4.379)

where $ f,\; g,$ and $ h $ are functions of temperature due to band parameters (see Kiehl and Ramanathan [93], and $ \overline{T} $ is an absorber mass-weighted mean temperature.

These path lengths are used extensively in the evaluation of $ A_{O_3}$ [137] and $ A_{CO_2}$ [87] and the trace gases. But path lengths dependent on both $ p^2$ (i.e. $ \xi$) and $ p$ (i.e. $ u$) are also needed in calculating the water-vapor absorptivity, $ \alpha_{H_2O}$ [138]. To account for the subdivided layer, a fractional layer amount must be multiplied by $ \xi$ and $ u$, e.g.

$\displaystyle \overline {\xi}_{11}$ $\displaystyle = \xi (p^k_H p^{k+1}_H) \times UINPL (1,k) ,$ (4.380)
$\displaystyle \overline {u}_{11}$ $\displaystyle = u(p^k_H, p^{k+1}_H) \times WINPL (1,k) ,$ (4.381)


$\displaystyle \overline {\beta}_{11}$ $\displaystyle = \beta (p^k_H, p^{k+1}_H) \times PINPL (1,k) ,$ (4.382)

where $ UINPL$, $ WINPL$, and $ PINPL$ are factors to account for the fractional subdivided layer amount. These quantities are derived for the case where the mixing ratio is assumed to be constant within a given layer (CO$ _2$ and H$ _2$O). For ozone, the mixing ratio is assumed to interpolate linearly in physical thickness; thus, another fractional layer amount $ ZINPL$ is required for evaluating $ A_{O_3}(p,p')$ across subdivided layers.

Consider the subdivided path for $ \alpha_{22}$; the total path length from $ p^k_H$ to $ p^{k+1}_H$ for the $ p^2$ path length will be

$\displaystyle \xi (p^k_H, p^{k+1}_H) \approx \overline{p}_H \left[p^k_H - p^{k+1}_H \right] ,$ (4.383)

where $ \overline {p}_H \equiv \frac{p^k_H + p^{k+1}_H}{2}$. The total layer path length is, therefore, proportional to

$\displaystyle \xi (p^k_H, p^{k+1}_H) \approx \frac{1}{2} (\left(p^{k}_{H}\right)^{2} - \left( p^{k+1}_{H}\right)^{2}).$ (4.384)

The path length $ \xi$ for $ \alpha_{22}$ requires the mean pressure

$\displaystyle \overline {p}_{22}$ $\displaystyle \approx \frac{1}{2} \left \{ \frac{p^k + p^{k+1}_H}{2} + p^{k+1}_H \right \} ,$ (4.385)

and the pressure difference

$\displaystyle \Delta p_{22}$ $\displaystyle \approx \frac{p^k + p^{k+1}_H}{2} - p^{k+1}_H .$ (4.386)

Therefore, the path $ \xi_{22}$ is

$\displaystyle \xi_{22} \approx \overline {p}_{22}\; \Delta p_{22} = \frac{1}{2}...
...t( \frac{p^k + p^{k+1}_H}{2} \right)^2 - \left( p^{k+1}_H \right)^2 \right \} .$ (4.387)

The fractional path length is obtained by normalizing this by $ \xi
(p^k_H, p^{k+1}_H)$,

$\displaystyle UINPL (2,k)$ $\displaystyle = DAF3(k) \left \{ \left( \frac{p^k + p^{k+1}_H}{2} \right )^2 - \left( p^{k+1}_H \right)^2 \right \} ,$ (4.388)


$\displaystyle DAF3(k)$ $\displaystyle = \frac{1}{\left( p^{k}_{H} \right)^2 - \left( p^{k+1}_{H}\right)^2} .$ (4.389)

Similar reasoning leads to the following expressions for the remaining fractional path lengths, for $ \alpha_{21}$,

$\displaystyle UINPL (3,k)$ $\displaystyle = DAF3(k) \left \{ \left( \frac{p^k + p^k_H}{2} \right)^2 - \left( p^{k+1}_{H} \right)^2 \right \} ,$ (4.390)

for $ \alpha_{11},$

$\displaystyle UINPL (1,k)$ $\displaystyle = DAF3(k) \left \{ \left(p^k_H \right)^2 - \left( \frac{p^k + p^k_H}{2} \right)^2 \right \} ,$ (4.391)

and for $ \alpha_{12}$,

$\displaystyle UINPL (4,k)$ $\displaystyle = DAF3(k) \left \{ \left(p^{k}_{H}\right)^2 - \left( \frac{p^k + p^{k+1}_H}{2} \right)^2 \right \} .$ (4.392)

The $ UINPL$ are fractional layer amounts for path length that scale as $ p^2$, i.e., $ \overline {\xi}_{ij}$.

For variables that scale linearly in $ p$, e.g. $ \overline {u}_{ij}$, the following fractional layer amounts are used:

$\displaystyle WINPL (1,k)$ $\displaystyle = DAF4 (k) \left\{ \frac{p^k_H - p^k}{2} \right \},$ (4.393)
$\displaystyle WINPL (2,k)$ $\displaystyle = DAF4 (k) \left \{ \frac{p^k - p^{k+1}_H}{2} \right \},$ (4.394)
$\displaystyle WINPL (3,k)$ $\displaystyle = DAF4 (k) \left \{ \left ( \frac{p^k_H + p^k}{2} \right) - p^{k+1}_H \right \},$ (4.395)
$\displaystyle WINPL (4,k)$ $\displaystyle = DAF4 (k) \left \{ p^k_H - \left ( {p^{k+1}_H + p^k}{2} \right ) \right \} ,$ (4.396)


$\displaystyle DAF4(k)$ $\displaystyle = \frac{1}{p^k_H - p^{k+1}_H} .$ (4.397)

These fractional layer amounts are directly analogous to the $ UINPL,$ but since $ \overline {u}$ is linear in $ p$, the squared terms are not present.

The variable $ \overline {\beta}_{ij}$ requires a mean pressure for the subdivided layer. These are

$\displaystyle PINPL (1,k)$ $\displaystyle = \frac{1}{2} \left \{ \frac{p^k + p^k_H}{2} + p^k_H \right \} ,$ (4.398)
$\displaystyle PINPL (2,k)$ $\displaystyle = \frac{1}{2} \left \{ \frac{p^k + p^{k+1}_H}{2} + p^{k+1}_h \right \} ,$ (4.399)
$\displaystyle PINPL (3,k)$ $\displaystyle = \frac{1}{2} \left \{ \frac{p^k + p^k_H}{2} + p^{k+1}_H \right \} ,$ (4.400)
$\displaystyle PINPL (4,k)$ $\displaystyle = \frac{1}{2} \left \{ \frac{p^k + p^{k+1}_H}{2} + p^k_H \right \} .$ (4.401)

Finally, fractional layer amounts for ozone path lengths are needed, since ozone is interpolated linearly in physical thickness. These are given by

$\displaystyle ZINPL (1,k)$ $\displaystyle = \frac{1}{2} \frac{\ln \left ( \frac{p^k_H}{p_k} \right )}{\ln \left ( \frac{p^k_H}{p^{k+1}_H} \right )} ,$ (4.402)
$\displaystyle ZINPL (2,k)$ $\displaystyle = \frac{1}{2} \frac{\ln \left ( \frac{p^k}{p^{k+1}_H} \right)} {\ln \left( \frac{p^k_H}{p^{k+1}_H} \right) },$ (4.403)
$\displaystyle ZINPL (3,k)$ $\displaystyle = ZINPL (1,k) + 2 ZINPL (2,k),$ (4.404)
$\displaystyle ZINPL (4,k)$ $\displaystyle = ZINPL (2,k) + 2 ZINPL (1,k) .$ (4.405)

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Next: 4.10 Surface Exchange Formulations Up: 4. Model Physics Previous: 4.8 Parameterization of Shortwave   Contents
Jim McCaa 2004-06-22