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4 . Radiation schemes in use at ECMWF
4.1 The operational longwave scheme
(as of March 2000)
The longwave radiation is an upgrade from
the original scheme defined by Morcrette et al. (1986), Morcrette (1990, 1991, 1993).
Assuming a non-scattering atmosphere in
local thermodynamic equilibrium,  is given by
where  is the monochromatic radiance at wavenumber
 at level  , propagating in a direction  (the angle that this direction makes with the vertical), where  and  is the monochromatic transmission through a layer whose limits are
at  and  seen under the same angle  , with  . The subscript `surf' refers to the earth's
surface.
After separating the upward and downward
components (indicated by superscripts + and -, respectively), and integrating
by parts, we obtain the radiation transfer equation as it is actually estimated
in the longwave part of the radiation code
where, taking benefit of the isotropic nature of the longwave
radiation, the radiance  of (5.3) has been replaced by the Planck function  in units of flux,  (here, and elsewhere,  is assumed to always includes the  factor).  is the surface temperature,  that of the air just above the surface,  is the temperature at pressure-level  ,  that at the top of the atmospheric model. The transmission
 is evaluated as the radiance transmission
in a direction  to the vertical such that  is the diffusivity
factor (Elsasser, 1942).
The integrals in (4.2) are evaluated numerically, after
discretization over the vertical grid, considering the atmosphere as a pile
of homogeneous layers. As the cooling rate is strongly dependent on local
conditions of temperature and pressure, and energy is mainly exchanged with
the layers adjacent to the level where fluxes are calculated, the contribution
of the distant layers is simply computed using a trapezoidal rule integration,
but the contribution of the adjacent layers is evaluated with a two-point
Gaussian quadrature, thus at the  level,
where  is the pressure corresponding to the Gaussian
root and  is the Gaussian weight.  and  are the Planck function gradients calculated between
two interfaces, and between mid-layer and interface, respectively.
The longwave spectrum is divided into six
spectral regions.
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1) 0 - 350  & 1450 - 1880  |
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2) 500 - 800  |
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3) 800 - 970  & 1110 - 1250  |
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4) 970 - 1110  |
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5) 350 - 500  |
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6) 1250 - 1450  & 1880 - 2820  |
corresponding to the centres of the rotation and vibration-rotation
bands of H2O, the 15  band of CO2, the atmospheric window, the 9.6  band of O3, the 25  "window" region, and the wings of the vibration-rotation band of
H2O, respectively.
Integration of (4.2) over wavenumber  within the  spectral region gives the upward and
downward fluxes as
The formulation accounts for the different temperature dependencies
involved in atmospheric flux calculations, namely that on  , the temperature at the level where fluxes are
calculated, and that on  , the temperature that governs the transmission through the temperature
dependence of the intensity and half-widths of the lines absorbing in the
concerned spectral region.
The band transmissivities are non-isothermal
accounting for the temperature dependence that arises from the wavenumber
integration of the product of the monochromatic absorption and the Planck
function. Two normalized band transmissivities are used for each absorber
in a given spectral region: the first one for calculating the first right-hand-side.
term in (4.2), involving the boundaries; it corresponds
to the weighted average of the transmission function by the Planck function
the second one for calculating the integral term in (4.2)
is the weighted average of the transmission function by the derivative of
the Planck function
where  is the pressure weighted amount of absorber.
In the scheme, the actual dependence on
 is carried out explicitly in the Planck functions
integrated over the spectral regions. Although normalized relative to  or  , the transmissivities still depend on  , both through Wien's displacement of the maximum of the Planck function
with temperature and through the temperature dependence of the absorption
coefficients. For computational efficiency, the transmissivities have been
developed into Pade approximants
where  is an effective amount of absorber which incorporates
the diffusivity factor  , the weighting of the absorber amount by pressure  , and the temperature dependence of the absorption coefficients.
The function  takes the form
The temperature dependence due to Wien's law is incorporated
although there is no explicit variation of the coefficients  and  with temperature. These coefficients have been computed for temperatures
between 187.5 and 312.5 K with a 12.5 K step, and transmissivities corresponding
to the reference temperature the closest to the pressure weighted temperature
 are actually used in the scheme.
The effect on absorption of the Doppler
broadening of the lines (important only for pressure lower than 10 hPa)
is included simply using the pressure correction method proposed by
Fels (1979) and incorporated by Giorgetta and Morcrette (1995).
The incorporation of the effects of clouds
on the longwave fluxes follows the treatment discussed by Washington
and Williamson (1977). Whatever the state of the cloudiness of the atmosphere,
the scheme starts by calculating the fluxes corresponding to a clear-sky
atmosphere and stores the terms of the energy exchange between the different
levels (the integrals in (5.4) Let  and  be the upward and downward clear-sky
fluxes. For any cloud layer actually present in the atmosphere, the scheme
then evaluates the fluxes assuming a unique overcast cloud of emissivity
unity. Let  and  the upward and downward fluxes when such a cloud
is present in the  layer of the atmosphere. Downward fluxes
above the cloud, and upward fluxes below the cloud, are assumed to be given
by the clear-sky values
Upward fluxes above the cloud ( for  ) and downward fluxes below it ( for  ) can be expressed with expressions similar to (4.3)
provided the boundary terms are now replaced by terms corresponding to possible
temperature discontinuities between the cloud and the surrounding air
where  is now the total Planck function (integrated
over the whole longwave spectrum) at level  , and  and  are the longwave fluxes at the upper and lower boundaries of the
cloud. Terms under the integrals correspond to exchange of energy between
layers in clear-sky atmosphere and have already been computed in the first
step of the calculations. This step is repeated for all cloudy layers. The
fluxes for the actual atmosphere (with semi-transparent, fractional and/or
multi-layered clouds) are derived from a linear combination of the fluxes
calculated in previous steps with some cloud overlap assumption in the case
of clouds present in several layers. Let  be the index of the layer containing the highest cloud,  the fractional cloud cover in layer  , with  for the upward flux at the surface, and with
 and  to have the right boundary condition for downward fluxes above the
highest cloud.
The maximum-random overlap assumption is
operationally used in the ECMWF model, and the cloudy upward  and downward  fluxes are obtained as
In case of semi-transparent clouds, the fractional cloudiness
entering the calculations is an effective cloud cover equal to the product
of the emissivity due to the condensed water and the gases in the layer
by the horizontal coverage of the cloud layer, with the emissivity,  , related to the condensed water amount
by
where  is the condensed water mass absorption coefficient
(in  ) following Smith and Shi (1992) for water clouds
and Ebert and Curry (1992)
for ice clouds.
4.2 The operational SW scheme
The rate of atmospheric heating by absorption
and scattering of shortwave radiation is
where  is the net total shortwave flux (the subscript
SW will be omitted in the remainder of this section).
is the diffuse radiance at wavenumber  , in a direction given by the azimuth angle,  , and the zenith angle,  , with  . In (4.15), 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
 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
 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.
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 Subsection 4.1). 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
and the flux averaged over the spectral interval  can then be calculated with the help of any band model of the transmission
function 
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 (4.19) (Fouquart,
1974). The main advantage of the method is that the actual distribution
 is smooth enough that (4.19) 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
where  is an absorption coefficient chosen to approximate
the spectrally averaged transmission of the clear sky atmosphere
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
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 (4.22) where  the amount of ozone is
 is the diffusivity factor (see Subsection 4.1), and  is the magnification factor (Rodgers, 1967) used instead of  to account for the sphericity of the atmosphere at very small
solar elevations
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.
4.2 (a) 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
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
where  is the cloud fractional coverage of the layer
within the cloudy fraction  of the column.
4.2 (b) 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
of the total single scattering albedo
of the total asymmetry factor
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
with  the effective total cloudiness over level 
and
 ,  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 (4.15) can be written as
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 (4.15) is integrated over  and  . The phase function is therefore given by
where  is the angle between incident and scattered
radiances. The integral in (4.15) thus becomes
where
is the asymmetry factor.
Using (4.33) in (4.15) after integrating over  and dividing by  , we get
We obtain a pair of equations for  and  by integrating (4.34) over 
For the cloudy layer assumed non-conservative ( ), the solutions to(4.35), for  , are
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
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
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
The solution of the Eddington's equations remains the same
provided that the total optical thickness, single scattering albedo and
asymmetry factor entering (4.36) to (4.38) take their transformed values
Practically, the optical thickness, single
scattering albedo, asymmetry factor and solar zenith angle entering (4.36)
to (4.38) are  ,  ,  and  defined in (4.39) and (4.40).
4.2 (c) 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
The optical thickness  of an atmospheric layer is simply
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
where  is the backscattering factor.
Practically,  and  are computed using (4.41) 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 (4.39) and (4.40))
As for their cloudy counterparts,  and  must account for the multiple reflections due to
the layers underneath
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 (4.24) and (4.41) attenuated by the gaseous transmissions
(4.22).
4.2 (d) 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  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, (4.17) should be written
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 (4.34)
and (4.40)) 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 (4.18) and starting from the surface
and working the formulas upward
where  and  are the layer reflectance and transmittance corresponding to a conservative
scattering medium.
Finally the upward and downward fluxes
are obtained as
4.2 (e) Cloud shortwave optical properties
As seen in (4.36), 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)
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
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  m.
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