Linearized Physics for Data Assimilation at ECMWF - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 256

F sw fluxes at a given level j are obtained from the reflectance and transmittance of

the atmospheric layers as

Y

F sw .j/ D F 0 T bot .k/ (11.12)

k D j

F sw .j/ D F sw .j/R top .j 1/

(11.13)

Computations of the transmittance at the bottom of a layer, T bot , start at the top

of atmosphere and work downwards. Those of the reflectance at the top of the
same layer,

R top , start at the surface and work upwards. In the presence of cloud
in the layer, the final fluxes are computed as a weighted average of the fluxes in the
clear-sky and in the cloudy fractions of the column (depending on the cloud-overlap
assumption).
The non-linear scheme is reasonably fast for application in 4D-Var and has
therefore been linearized without a-priori changes ( Janiskova et al. 2002 ). The only
modification with respect to the non-linear version (used operationaly until June
2007; since then Rapid Radiation Transfer model for SW radiation is used - Mlawer
and Clough 1997 ), is the use of two spectral intervals, instead of six intervals. This
is meant to reduce the computational cost.

The Longwave Radiation Scheme

The LW radiation scheme, used in the ECMWF full NL forecast model is the Rapid
Radiation Transfer Model (RRTM; Mlawer et al. 1997 ; Morcrette et al. 2001 ).
The complexity of the RRTM scheme for the LW part of the spectrum makes
accurate computations expensive. In the variational assimilation framework, the
older operational scheme of Morcrette ( 1989 ) was linearized. In this scheme, the LW
spectrum from 0 to 2,820 cm 1 is divided into six spectral regions. The transmission
functions for water vapour and carbon dioxide over those spectral intervals are fitted
using Pade approximations on narrow-band transmissions obtained with statistical
band models ( Morcrette et al. 1986 ). Integration of the radiation transfer equation
over wavenumber within the particular spectral regions yields the upward and

downward fluxes.
The inclusion of cloud effects on the LW fluxes follows the treatment discussed
by Washington and Williamson ( 1997 ). The scheme first calculates upward and
downward fluxes ( F 0 .i/ F 0 .i/ ) for a clear-sky atmosphere. In any cloudy layer,

the scheme evaluates the fluxes assuming a unique overcast cloud of emissivity
unity, i.e.

and

F n .i/ F n .i/ th layer of the atmosphere.

The fluxes for the actual atmosphere are derived from a linear combination of the
fluxes calculated in the previous steps with some cloud overlap assumption (see
below) in the case of clouds spreading over several layers. If

and for a cloud present in the n

N is the number of

model layers starting from the top of atmosphere to the bottom, C i the fractional