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Chapter 2. Radiation
IFS documentation Front Page
Chapter 1. Overview
Chapter 2. Radiation
Chapter 3. Turbulent diffusion and interactions with the surface
Chapter 4. Subgrid-scale orographic drag
Chapter 5. Convection
Chapter 6. Clouds and large-scale precipitation
Chapter 7. Land suface parametrization
Chapter 8. Methane oxidation
Chapter 9. Climatological data
REFERENCES
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2.3 Shortwave Radiation
The rate of atmospheric heating by absorption and scattering of shortwave radiation is
|
(2.19) |
where
is the net total shortwave flux (the subscript SW will be omitted
in the remainder of this section).
|
(2.20) |
is the diffuse radiance at wavenumber
, in a direction
given by the azimuth angle, 
, and the zenith
angle, 
, with 
. In (2.20),
we assume a plane parallel atmosphere, and the vertical coordinate is the
optical depth 
, a convenient variable when the energy source is outside
the medium
|
(2.21) |
is the extinction coefficient, equal to the sum of the scattering
coefficient 
of the aerosol (or cloud particle absorption
coefficient 
) and the purely molecular absorption
coefficient 
. The diffuse radiance 
is governed by the radiation transfer equation
|
(2.22) |
is the incident solar irradiance in the direction 
, 
is the single scattering albedo (
) and 
is the scattering phase function which defines the probability that
radiation coming from direction (
) is scattered
in direction (
). The shortwave part of the scheme, originally
developed by Fouquart and Bonnel (1980) solves
the radiation transfer equation and integrates the fluxes over the whole
shortwave spectrum between 0.2 and 4 
. Upward and downward fluxes are obtained from the reflectance
and transmittances of the layers, and the photon-path-distribution method
allows to separate the parametrization of the scattering processes from
that of the molecular absorption.2.3.1 Spectral integration
Solar radiation is attenuated by absorbing gases, mainly water vapour, uniformly mixed gases (oxygen, carbon dioxide, methane, nitrous oxide) and ozone, and scattered by molecules (Rayleigh scattering), aerosols and cloud particles. Since scattering and molecular absorption occur simultaneously, the exact amount of absorber along the photon path length is unknown, and band models of the transmission function cannot be used directly as in longwave radiation transfer (see Section 2.2). The approach of the photon path distribution method is to calculate the probability
that a photon
contributing to the flux 
in the conservative
case (i.e., no absorption, 
, 
) has encountered an absorber amount between 
and 
.With this distribution, the radiative flux at wavenumber 
is related to 
by
|
(2.23) |
and the flux averaged over the spectral interval
can then be
calculated with the help of any band model of the transmission function
|
(2.24) |
To find the distribution function
, the scattering problem is solved first,
by any method, for a set of arbitrarily fixed absorption coefficients 
, thus giving a set of simulated fluxes 
. An inverse Laplace transform is then performed on (2.23)
(Fouquart, 1974). The main advantage
of the method is that the actual distribution 
is smooth enough
that (2.23) gives accurate results
even if 
itself is not
known accurately. In fact, 
need not be
calculated explicitly as the spectrally integrated fluxes are
where
and 
.The atmospheric absorption in the water vapour bands is generally strong, and the scheme determines an effective absorber amount
between 
and 
derived from
|
(2.25) |
where
is an absorption coefficient chosen to approximate the spectrally
averaged transmission of the clear sky atmosphere
|
(2.26) |
where
is the total amount of absorber in a vertical column and 
. Once the effective absorber amounts of 
and uniformly mixed gases are found, the transmission functions are
computed using Pade approximants
|
(2.27) |
Absorption by ozone is also taken into account, but since ozone is located at low pressure levels for which molecular scattering is small and Mie scattering is negligible, interactions between scattering processes and ozone absorption are neglected. Transmission through ozone is computed using (2.24) where
the amount
of ozone is
is the diffusivity factor (see Section 2.2), and 
is the magnification
factor (Rodgers, 1967) used instead of 
to account for the sphericity of the atmosphere at very small
solar elevations
|
(2.28) |
To perform the spectral integration, it is convenient to discretize the solar spectral interval into subintervals in which the surface reflectance can be considered as constant. Since the main cause of the important spectral variation of the surface albedo is the sharp increase in the reflectivity of the vegetation in the near infrared, and since water vapour does not absorb below 0.68
, the shortwave scheme considers two spectral intervals, one for
the visible (0.2-0.68 
), one for
the near infrared (0.68-4.0 
) parts of
the solar spectrum. This cut-off at 0.68 
also makes
the scheme more computationally efficient, in as much as the interactions
between gaseous absorption (by water vapour and uniformly mixed gases) and
scattering processes are accounted for only in the near-infrared interval.2.3.2 Vertical integration
Considering an atmosphere where a fraction
(as seen from
the surface or the top of the atmosphere) is covered by clouds (the fraction
depends on which cloud-overlap assumption is
assumed for the calculations), the final fluxes are given as a weighted
average of the fluxes in the clear sky and in the cloudy fractions of the
column
where the subscripts `
' and `
' refer to
the clear-sky and cloudy fractions of the layer, respectively. In contrast
to the scheme of Geleyn and Hollingsworth (1979),
the fluxes are not obtained through the solution of a system of linear equations
in a matrix form. Rather, assuming an atmosphere divided into homogeneous
layers, the upward and downward fluxes at a given layer interface 
are given by
|
(2.29) |
where
and 
are the reflectance at the top and the transmittance
at the bottom of the 
th layer. Computations
of 
's start at the surface and work upward, whereas
those of 
's start at the top of the atmosphere and work
downward. 
and 
account for
the presence of cloud in the layer
|
(2.30) |
where
is the cloud fractional coverage of the layer within the cloudy
fraction 
of the column.2.3.2 (a) Cloudy fraction of the layer
and 
are the reflectance at the top and transmittance
at the bottom of the cloudy fraction of the layer calculated with the Delta-Eddington
approximation. Given 
, 
, and 
, the optical thicknesses for the cloud, the aerosol
and the molecular absorption of the gases, respectively, and (
), and 
and 
the cloud and aerosol asymmetry factors, 
and 
are calculated as functions of the total optical thickness of the
layer
|
(2.31) |
of the total single scattering albedo
|
(2.32) |
of the total asymmetry factor
|
(2.33) |
of the reflectance
of the underlying medium (surface or layers below
the 
th interface), and of the cosine of an effective
solar zenith angle 
which accounts for the decrease of the
direct solar beam and the corresponding increase of the diffuse part of
the downward radiation by the upper scattering layers
|
(2.34) |
with
the effective total cloudiness over level 
|
(2.35) |
and
|
(2.36) |
, 
and 
are the optical thickness, single scattering
albedo and asymmetry factor of the cloud in the 
th layer, and 
is the diffusivity
factor. The scheme follows the Eddington approximation first proposed by
Shettle and Weinman (1970), then
modified by Joseph et al. (1976) to account more
accurately for the large fraction of radiation directly transmitted in the
forward scattering peak in case of highly asymmetric phase functions. Eddington's
approximation assumes that, in a scattering medium of optical thickness
, of single
scattering albedo 
, and of asymmetry factor 
, the radiance 
entering (2.17) can be written as
|
(2.37) |
In that case, when the phase function is expanded as a series of associated Legendre functions, all terms of order greater than one vanish when (2.20) is integrated over
and 
. The phase function is therefore given
by
where
is the angle between incident and scattered radiances. The
integral in (2.20) thus becomes
|
(2.38) |
where
is the asymmetry factor.
Using (2.38) in (2.20) after integrating over
and dividing by 
, we get
|
(2.39) |
We obtain a pair of equations for
and 
by integrating
(2.39) over 
|
(2.40) |
For the cloudy layer assumed non-conservative (
), the solutions
to (2.39) and (2.40),
for 
, are
|
(2.41) |
where
The two boundary conditions allow to solve the system for
and 
; the downward directed diffuse flux at the top of the atmosphere
is zero, i.e.,
which translates into
|
(2.42) |
The upward directed flux at the bottom of the layer is equal to the product of the downward directed diffuse and direct fluxes and the corresponding diffuse and direct reflectance (
and 
, respectively) of the underlying medium
which translates into
|
(2.43) |
In the Delta-Eddington approximation, the phase function is approximated by a Dirac delta function forward-scatter peak and a two-term expansion of the phase function
where
is the fractional scattering into the forward peak and 
the asymmetry factor of the truncated phase function. As shown
by Joseph et al.
(1976), these parameters are
|
(2.44) |
The solution of the Eddington's equations remains the same provided that the total optical thickness, single scattering albedo and asymmetry factor entering (2.39)-(2.43) take their transformed values
|
(2.45) |
Practically, the optical thickness, single scattering albedo, asymmetry factor and solar zenith angle entering (2.39)-(2.42) are
, 
, 
and 
defined in (2.33)
and (2.34).2.3.2 (b) Clear-sky fraction of the layers
In the clear-sky part of the atmosphere, the shortwave scheme accounts for scattering and absorption by molecules and aerosols. The following calculations are practically done twice, once for the clear-sky fraction (
) of the atmospheric
column with 
equal to 
, simply modified
for the effect of Rayleigh and aerosol scattering, the second time for the
clear-sky fraction of each individual layer within the fraction 
of the atmospheric column containing clouds, with
equal to 
.As the optical thickness for both Rayleigh and aerosol scattering is small,
and 
, the reflectance at the top and transmittance
at the bottom of the 
th layer can
be calculated using respectively a first and a second-order expansion of
the analytical solutions of the two-stream equations similar to that of
Coakley and Chylek (1975). For
Rayleigh scattering, the optical thickness, single scattering albedo and
asymmetry factor are respectively 
, 
, and 
, so that
|
(2.46) |
The optical thickness
of an atmospheric layer is simply
|
(2.47) |
where
is the Rayleigh optical thickness of the whole atmosphere parametrized
as a function of the solar zenith angle (Deschamps
et al., 1983)For aerosol scattering and absorption, the optical thickness, single scattering albedo and asymmetry factor are respectively
, 
, with 
and 
, so that
|
(2.48) |
|
(2.49) |
where
is the backscattering factor.Practically,
and 
are computed using (2.49) and the combined effect of aerosol
and Rayleigh scattering comes from using modified parameters corresponding
to the addition of the two scatterers with provision for the highly asymmetric
aerosol phase function through Delta-approximation of the forward scattering
peak (as in (2.40)-(2.41))
|
(2.50) |
As for their cloudy counterparts,
and 
must account
for the multiple reflections due to the layers underneath
|
(2.51) |
and
is the reflectance of the underlying medium 
and 
is the diffusivity factor.Since interactions between molecular absorption and Rayleigh and aerosol scattering are negligible, the radiative fluxes in a clear-sky atmosphere are simply those calculated from (2.27) and (2.45) attenuated by the gaseous transmissions (2.25).
2.3.3 Multiple reflections between layers
To deal properly with the multiple reflections between the surface and the cloud layers, it should be necessary to separate the contribution of each individual reflecting surface to the layer reflectance and transmittances in as much as each such surface gives rise to a particular distribution of absorber amount. In case of an atmosphere including N cloud layers, the reflected light above the highest cloud consists of photons directly reflected by the highest cloud without interaction with the underlying atmosphere, and of photons that have passed through this cloud layer and undergone at least one reflection on the underlying atmosphere. In fact, (2.22) should be written
|
(2.52) |
where
and 
are the conservative fluxes and the distributions
of absorber amount corresponding to the different reflecting surfaces.Fouquart and Bonnel (1980) have shown that a very good approximation to this problem is obtained by evaluating the reflectance and transmittance of each layer (using (2.39) and (2.45)) assuming successively a non-reflecting underlying medium (
), then a reflecting underlying medium
(
). First calculations provide the contribution to
reflectance and transmittance of those photons interacting only with the
layer into consideration, whereas the second ones give the contribution
of the photons with interactions also outside the layer itself.From those two sets of layer reflectance and transmittances (
) and (
) respectively, effective absorber amounts to be
applied to computing the transmission functions for upward and downward
fluxes are then derived using (2.23) and starting from the surface and working
the formulas upward
|
(2.53) |
where
and 
are the layer reflectance and transmittance
corresponding to a conservative scattering medium.Finally the upward and downward fluxes are obtained as
|
(2.54) |
|
(2.55) |
2.3.4 Cloud shortwave optical properties
As seen in Sub-section 2.3.2 (a), the cloud radiative properties depend on three different parameters: the optical thickness
, the asymmetry factor 
, and the single
scattering albedo 
. Presently the cloud optical properties are derived from Fouquart (1987) for the water clouds, and Ebert and Curry (1992) for the ice clouds
is related to the cloud liquid water amount 
by
where
is the mean effective radius of the size distribution of the
cloud water droplets. Presently 
is parametrized
as a linear function of height from 10 
m at the surface
to 45 
m at the top of the atmosphere, in an empirical
attempt at dealing with the variation of water cloud type with height. Smaller
water droplets are observed in low-level stratiform clouds whereas larger
droplets are found in mid-level cumuliform water clouds.In the two spectral intervals of the shortwave radiation scheme,
is fixed to 0.865 and 0.910, respectively, and 
is given as a function of 
following
Fouquart (1987)
|
(2.56) |
These cloud shortwave radiative parameters have been fitted to in situ measurements of stratocumulus clouds (Bonnel et al., 1983).
For the optical properties of ice clouds, we have
|
(2.57) |
where the coefficients have been derived from Ebert and Curry (1992) for the two intervals of the shortwave radiation scheme, and
is fixed at 40 
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05.04.2002 |
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