Estimation of the uncertainty in the complete (i.e. the first and second indirect
effects) radiative forcing is not currently feasible due to a lack of analytical
relationships to treat the indirect forcing of the second kind. However, the
indirect forcing of the first kind can be treated if we adopt a simple box-model
approach such as those used in early assessments of indirect forcing (e.g.,
Charlson et al., 1992; Schwartz and Slingo, 1996).This evaluation of uncertainty
is only illustrative both because of the box-model nature of the estimate and
because our assessment of the uncertainty in the parameters is only first order.
Nevertheless, it is useful to make such calculations since they can yield valuable
information both on our current state of knowledge with regard to the indirect
forcing and can help guide efforts to reduce uncertainty. Moreover they illustrate
a rigorous method that could allow a more quantitative estimate of uncertainty
than the methods followed in Chapter 6.

We adopt a functional relationship between sulphate concentrations and cloud
droplet number concentration based on empirical relationships in order to render
the calculations more tractable. Therefore, this analysis is only applicable
to the Northern Hemisphere, since data from this region were used to derive
the empirical relationship. The analysis is further restricted to the marine
atmosphere and excludes any consideration of indirect forcing by biomass aerosols.

The indirect forcing of the first kind ( hereafter called simply the indirect forcing) can be expressed as:

where F_{d} is the average downward flux at the top of the atmosphere,
T_{a} is the atmospheric transmission above the cloud layer, f_{c}
is the fractional cloud cover of those clouds susceptible to aerosol modulation
and DA_{p} is the change in planetary albedo
(equivalent here to the above cloud albedo) associated with an increase in the
cloud droplet number concentration (CDNC). To take into account multiple reflections
between the cloud layer and the surface, the expression of Liou (1980) is used
with an assumption of no absorption within the cloud. This assumption is very
reasonable for the bulk of the incoming radiation which will be scattered by
cloud drops, i.e., subject to the indirect effect, but does restrict the radiation
band to a range from 0.3 to 0.7 mm. Thus:

where A_{c} is the cloud albedo and R_{s} is the albedo of
the underlying surface. DA_{p} is then calculated
as the difference between this function evaluated for A_{c}, the background
cloud albedo, and A_{c}', the anthropogenically perturbed albedo (note
that "primed" quantities will always refer to anthropogenically perturbed
values of the quantity). Cloud albedo is, in turn, evaluated using the relationship:

which is an approximation of the two stream evaluation of cloud albedo for
conservative scattering assuming an asymmetry parameter of 0.85 (Lacis and Hansen,
1974). Here, t_{c} is the cloud optical depth,
given by the expression of Twomey (1977):

where h is the cloud layer thickness, LWC is the layer mean liquid-water content,
N_{d} is the CDNC, and r is the density of
water. To relate the CDNC to anthropogenic emissions, we use the empirical expression
of Boucher and Lohmann (1995) which has the form:

where SO_{4}^{2-} is the mean concentration of sulphate aerosol
at cloud base in µg m^{-3}, and A and B are empirical constants.
We adopt the values of A=115 and B=0.48 which are appropriate for marine air
(Boucher and Lohmann, 1995).

Using the same procedures as with the assessment of uncertainties in the direct
forcing (e.g. Section 5.4.2), we first determine the
uncertainties in the most fundamental parameters. We then use Taylor expansions
of the various equations given above to determine the uncertainty in the forcing
associated with the central values of the parameters used in the calculations.
Thus the uncertainty involves the uncertainties in the concentration of SO_{4}^{2-}
and in the empirical coefficients used to relate the concentration of SO_{4}^{2-}
to N^{d}. Moreover, the calculation of uncertainty necessarily involves
an evaluation of A_{c} for both the background and anthropogenically
perturbed values. These quantities, together with corresponding values for LWC
and h, which are based on available observations, are then used to generate
uncertainties in the cloud optical depth. This hierarchical evaluation proceeds
"upward" from the uncertainty in the primary variables until the uncertainty
in the forcing itself, together with the associated central value can be assessed.
Such an uncertainty estimate is different in philosophy from that used in Chapter
6 which only assesses the range of estimates in the literature.

*Continues on next page*

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