- 4.9.1 Major absorbers
- 4.9.2 Water vapor
- 4.9.3 Trace gas parameterizations
- 4.9.4 Mixing ratio of trace gases
- 4.9.5 Cloud emissivity
- 4.9.6 Numerical algorithms and cloud overlap

where is the Stefan-Boltzmann relation. The pressures and refer to the top of the model and the surface, respectively. and are the absorptivity and emissivity

where the integration is over wavenumber . is the Planck function, and is the atmospheric transmission. Thus, to solve for fluxes at each model layer we need solutions to the following:

where 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 , and hence can be evaluated at the band center, i.e.

where is the band absorptance (or equivalent width) in units of cm. Note that , 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 :

where is an empirical constant. is the scaled dimensionless path length

where is the band strength, is the mass mixing ratio of the absorber, and is the density of air. is a line width factor,

where is the mean line halfwidth for the band, is the atmospheric pressure, is a reference pressure, and is the mean line spacing for the band. The determination of , , 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, O, CH, NO, and minor absorption bands of CO, while for the CFCs and stratospheric aerosols we employ the exponential transmission approximation discussed by Ramanathan et al. [139]

where is the band width, and is the absorber path length

and 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, the radiative flux is determined in part by the integral

which can be re-formulated for given sub intervals in wavenumber as

The factors represent the transmissions through stratospheric volcanic aerosols. The transmissions in each band are replaced by effective transmissions given by:

(4.242) |

where is the diffusivity factor, is an effective specific extinction for the band, and 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 band of NO is

(4.243) |

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.

Details of the parameterization for the three major absorbers, HO, CO and O, 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 HO by Ramanathan and Downey [138] has been replaced a new formulation in CAM 3.0.

For CO

is evaluated for cm, where is the broad-band absorptance from Kiehl and Briegleb [87]. Similarly,

For ozone,

where is the ozone broad-band absorptance from Ramanathan and Dickinson [137]. The longwave absorptance formulation includes a Voigt line profile effects for CO and O. For the mid-to-upper stratosphere ( 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, ,

where for CO and for . These values insure agreement with line-by-line cooling rate calculations up to mb.

4.9.2 Water vapor

Water vapor cannot employ the broad-band absorptance method since HO 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 HO. In this version of GENLN3, the parameters for HO 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 HO lines alone and for HO 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, and the non-window is the remainder of the spectrum between 20 to 2200 cm. Outside the mid-infrared window (the so-called non-window region), the HO 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:

(4.250) |

where is the window component and is the non-window component for the portion of the atmosphere bounded by pressures and .

Let represent the total non-window absorption for a homogeneous atmosphere characterized by a set of scaling parameters . 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

In the following expressions, a tilde denotes a parameter derived using scaling theory for the equivalence between homogeneous and inhomogeneous atmospheres. The subscript 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:

(4.252) |

Here is the pressure-weighted precipitable water, is the scaled atmospheric pressure, is the emission temperature of radiation, is the absorber weighted path temperature, and is the scaled relative humidity. The subscript indicates that the quantities are evaluated for the non-window.

The absorber-weighted path temperature is:

where is the thermodynamic temperature of the atmosphere at pressure . The HO path or precipitable water is:

(4.254) | |||

where is the specific humidity at pressure and is the acceleration of gravity. The HO path and pressure for a homogeneous atmosphere with equivalent line absorption are [62]

where

(4.257) | |||

(4.258) |

The factor is the line strength for each line in the spectral interval under consideration. The characteristic width of each line at a reference pressure and specific humidity is . It is convenient to calculate the absorptance in terms of a pressure-weighted HO path

(4.259) |

The equivalent pressure-weighted HO path is simply

Although the relative humidity (or HO 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
is defined in terms of an
effective HO specific humidity
and saturation specific
humidity
along the path:

where is the saturation vapor pressure at temperature , is an effective pressure, and is the ratio of gas constants for air and water vapor.

The window term
requires a special provision for the different
path parameters for the lines and continuum. Let

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

(4.266) |

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

(4.267) |

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

where is a reference temperature, is a suitably chosen wavenumber inside the window, is the self-continuum path length, and is the self continuum absorption coefficient. The self-continuum path length may be approximated by

The lines-only absorptivity can be written in terms of a line transmission factor and an asymptotic absorptivity in the limit of a black-body atmosphere. is a function only of [138]. The relationship is

Define an effective continuum transmission by setting

We approximate the window absorptivity by:

(4.272) |

This approximation for 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:

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

(4.274) |

The volcanic transmission factor is

where is the diffusivity factor, is an effective specific extinction for the band, and is the mass path of the volcanic aerosols. The extinction 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).

**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

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,

where according to (4.235),

where is a path weighted temperature,

The dimensionless path length is,

and the mean line width factor is,

where is the mass mixing ratio of methane, is the local layer temperature in Kelvin and is the pressure in Pascals, and is Pa. is a diffusivity factor of 1.66. The water vapor overlap factor for this spectral region is,

and 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 bands. The
radiative effects of the 1285 cm band is given by the last term
in (4.276),

The absorptance for the 1285 cm NO band is given by

where , account for the fundamental transition, while , account for the first ``hot'' band transition. These parameters are defined as

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

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].

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

where and 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 NO band.

The 1168 cm NO band system is represented by the seventh
term on the RHS of (4.241). This term can be re-written as

where the last term accounts for the 1168 cm NO band. For the broad band formulation this expression becomes,

The band absorptance for the 1168 cm NO band is given by

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

Note that the 1168 cm 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,

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

where and are coefficients that are obtained by fitting (4.298) to the averaged transmission from a 10 cm narrow band Malkmus. The path length is,

where and account for the temperature dependence of the spectroscopic parameters [151]

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

The p-type continuum is represented by

The factors , , and are listed for specific spectral intervals in Table 4.4.

Index | |||||

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 |

Index | |||||

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 NO band centered at 589 cm is represented by the
first term on the RHS of (4.241),

where the last term in (4.307) represents the radiative effects of the 589 cm NO band,

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

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

The overlap effect of water vapor is given by the transmission factor for the 500 to 800 cm 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 bands near 589 cm is obtained from the formulation in Kiehl and Briegleb [87],

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

**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

where is the width of the CFC absorption band, is the band strength, is the abundance of CFC (g cm),

where is the mass mixing ratio of either CFC11 or CFC12. is the diffusivity factor. In the linear limit , since (4.317) deviates slightly from the pure linear limit we let . 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).

For the 798 cm CFC11 band, the absorption effect is given by the second term on the right hand side of (4.319),

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 CFC11 band is represented by the second term on the RHS of (4.320),

where the HO overlap factor is given by index 2 in Tables 4.3 and 4.4. The 933 cm CFC11 band is given by the third term on the RHS of (4.322),

where the HO 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 is represented by the fourth term on the RHS of (4.323),

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

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

For the 889 cm CFC12 band the absorption is defined by the second term in (4.321) as

where the HO 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 CFC12 band is described by the second term in (4.322),

where the HO overlap is defined as index 4 in Tables 4.3 and 4.4. The 1102 cm CFC12 band is represented by the last term on the RHS of (4.323),

where the transmission by ozone is described by (4.328) and the HO overlap factor is represented by index 5 in Tables 4.3 and 4.4. The final CFC12 band at 1161 cm is represented by the second term on the RHS of (4.292),

where the HO overlap factor is defined as index 6 in Tables 4.3 and 4.4.

**Minor CO 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, such as during the Archean. The first band
we consider is centered at 961 cm. The radiative contribution
of this band is represented by the last term in (4.322),

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

where the path length parameters are defined as

and the pressure parameter is,

and,

The CO band centered at 1064 cm is represented by the third term on the RHS of (4.323),

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 CO band is given by

where the dimensionless path length is defined as

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

In the above expressions, is the column mass abundance of CO,

(4.350) |

where is the mass mixing ratio of CO (assumed constant).

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,

where denotes the volume mixing ratio of these gases. The CAM 3.0 employs volume mixing ratios for the year 1992 based on IPCC [79], , , and . The pressure level (mb) of the tropopause is defined as

For , the stratospheric mixing ratios are defined as

where the mixing ratio scale heights are defined as

and,

where is latitude in degrees.

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

where is a diffusivity factor set to 1.66, is the longwave absorption coefficient ( ), and CWP is the cloud water path (). The absorption coefficient is defined as

where is the longwave absorption coefficient for liquid cloud water and has a value of 0.090361, such that is 0.15. 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,

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 . 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 ( ). 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., ) using the emissivity array. However, the flux boundary conditions and 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 and . 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 within a
maximum-overlap region are given by:

where and are the upward and downwelling fluxes for the cloud configuration . The symbols required to write these fluxes are defined in Table 4.6.

Stefan-Boltzmann constant | |

pressure | |

pressure at top of layer | |

pressure at bottom of layer ( ) | |

temperature at pressure | |

layer containing pseudo cloud for b.c. | |

layer containing pseudo cloud for b.c. | |

emissivity of cloud in layer | |

emissivity of pseudo clouds at and | |

clear-sky absorptivity from pressure to | |

downwelling clear-sky flux at layer | |

upwelling clear-sky flux at layer | |

weights for up/downwelling clear-sky flux at layer | |

weights for up/downwelling flux at layer from cloud at |

The downward and upward fluxes for each configuration can be derived
by iterating the longwave equations from TOA and the surface to the
layer . At each iteration, the solutions are advanced between
successive cloud layers. The final form of the fluxes in
configuration
is:

The clear-sky and cloudy-sky weights are:

The longwave atmospheric heating rate is obtained from

which is added to the nonlinear term 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 and emissivity . To reduce this computational burden, and 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 and 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 in Equations (4.229)-(4.230). The reason for the additional care in evaluating this integral arises from the nonlinear behavior of across a given model layer. For example, if the flux at interface is required, an integral of the form must be evaluated. For the nearest layer to level , the following terms will arise:

employing the trapezoidal rule. The problem arises with the second absorptivity , since this term is zero. It is also known that is nearly exponential in form within a layer. Thus, to accurately account for the variation of across a layer, many more grid points are required than are available in CAM 3.0. The nearest layer must, therefore, be subdivided and must be evaluated across the subdivided layers. The algorithm that is employed in is to use a trapezoid method for all layers

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

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

The
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 [*e.g.,*
Eq. (4.373)]. Temperatures are known at model levels.
Temperatures at layer interfaces are determined through linear
interpolation in between layer midpoint temperatures. Thus,
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:

where and are functions of temperature due to band parameters (see Kiehl and Ramanathan [93], and is an absorber mass-weighted mean temperature.

These path lengths are used extensively in the evaluation of
[137] and [87] and the trace
gases. But path lengths dependent on both (*i.e.* ) and
(*i.e.* ) are also needed in calculating the water-vapor
absorptivity,
[138]. To account for
the subdivided layer, a fractional layer amount must be multiplied by
and , *e.g.*

where , , and 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 and HO). For ozone, the mixing ratio is assumed to interpolate linearly in physical thickness; thus, another fractional layer amount is required for evaluating across subdivided layers.

Consider the subdivided path for ; the total path length from to for the path length will be

where . The

The path length for requires the mean pressure

Therefore, the path is

The fractional path length is obtained by normalizing this by ,

Similar reasoning leads to the following expressions for the remaining fractional path lengths, for ,

The are fractional layer amounts for path length that scale as ,

For variables that scale linearly in , *e.g.*
,
the following fractional layer amounts are used:

These fractional layer amounts are directly analogous to the but since is linear in , the squared terms are not present.

The variable requires a mean pressure for the subdivided layer. These are

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