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  and Kiehl and Ramanathan . This approach is based on the earlier work of Ramanathan . 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.
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 ). 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. , Kiehl and Briegleb , and Ramanathan and Dickinson , 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  has been replaced a new formulation in CAM 3.0.
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.  is used for water-vapor absorptivities and emissivities. This parameterization replaces the scheme developed by Ramanathan and Downey  used in previous versions of the model. The new formulation uses the line-by-line radiative transfer model GENLN3  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 , and the continuum is treated with the Clough, Kneizys, and Davies (CKD) model version 2.4.1 . 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 .
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 . 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:
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 . 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 , 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:
The absorber-weighted path temperature is:
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
along the path:
The window term
requires a special provision for the different
path parameters for the lines and continuum. Let
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 . The relationship is
In the presence of stratospheric volcanic aerosols, the
expressions for the absorptivity become:
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
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 1168 cm NO band system is represented by the seventh
term on the RHS of (4.241). This term can be re-written as
|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|
|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),
CFCs. The effects of both CFC11 and CFC12 are included by using the approach of Ramanathan et al. . Thus, the band absorptance of the CFCs is given by
The contribution by these CFC absorption bands is accounted for by the following terms in (4.241).
|Band Number||Band Center|
For the 798 cm CFC11 band, the absorption effect is given by the second term on the right hand side of (4.319),
For the 889 cm CFC12 band the absorption is defined by the second term in (4.321) as
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),
The pressure factor, , for (4.343) is the same as
defined in (4.338), while the other factors are,
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,
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
The treatment of cloud overlap follows Collins . 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  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 . 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 . 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:
|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
The longwave atmospheric heating rate is obtained from
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:
For the upward flux, the nearest layer contribution to the integral is evaluated from
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:
These path lengths are used extensively in the evaluation of  and  and the trace gases. But path lengths dependent on both (i.e. ) and (i.e. ) are also needed in calculating the water-vapor absorptivity, . To account for the subdivided layer, a fractional layer amount must be multiplied by and , e.g.
Consider the subdivided path for ; the total path length from to for the path length will be
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