Several issues arise that deserve some discussion. The first such issue is the concatenation of uncertainties in the empirical relationship between the concentration of SO₄^2- andN_d. The uncertainty in SO₄^2- is straightforward and is based on the assessment of uncertainty in burden and the uncertainty in emissions as used in Section 5.4.2. The uncertainty in the relationship between SO₄^2- andN_d, equation (3), requires an evaluation of the uncertainties in the coefficients A and B. Based on a comparison of the parametrization of Boucher and Lohmann (1995) with that of Jones et al. (1994b) (Figure 5.7), we assign an ad hoc contribution of the functional relationship to the uncertainty inN_dof 40% of the central value. We then further assume that the uncertainties in A and B contribute equally to the total uncertainty (i.e., 40%) and, with these constraints, derived the uncertainties in A and B. This allows us to use Taylor expansions for both the perturbed and unperturbed atmosphere. Due largely to the form of the functional relationship, this procedure yields uncertainties whose major components in both the perturbed and unperturbed cases are attributable to the uncertainty in the parametrization (61% for the unperturbed case and 64% for the perturbed case) rather than the uncertainty in burden or emissions.

In order to assess the overall uncertainty in forcing, the covariance between the base parametersN_d, LWC and h must be evaluated. But both LWC and h are assumed to be constant in the first indirect effect, and therefore the covariance is assumed to be zero. Certainly, the limited observations available (cf., Hegg et al., 1996b) do not show any covariance. Nevertheless, it is important to note this potential effect and the possible impact of the second indirect effect (precipitation modulation effects by aerosols) on the uncertainty in the first.

Another covariance which cannot be neglected arises in the evaluation of the uncertainty in DAp from the possible covariance between A_c and A_c'. This covariance arises because of the dependence of the perturbation in cloud albedo on both the unperturbed aerosol concentration and the unperturbed albedo. We assessed this covariance by using equations (4) and (3) to generate a set of corresponding values of the perturbed and unperturbed albedos for different values ofN_d. Then a linear fit to the parameters DN_d and A_c' was used to derive the correlation coefficient and then the covariance. Various sample sizes and incremental values of the aerosol perturbation were generated to test the stability of the correlation. This procedure resulted in a stable correlation coefficient between A_c and A_c' of 0.74.

A final issue arises as to the choice of susceptible cloud fraction (f_c) in the basic forcing equation. Here, we follow the analysis of Charlson et al. (1987) and use the estimates in the cloud atlas of Warren et al. (1988). The total fractional cover of low and mid level clouds is not used since mid-level clouds can be mixed phase and the relationship of these clouds to anthropogenic aerosol is still unclear (Section 5.3.6). Instead, we use the estimates of Charlson et al. (1987) for non-overlapped low marine cloud as a lower bound (2/3 bound) for the susceptible cloud fraction and the sum (correcting for overlap) of the low and middle marine stratiform cloud as an upper limit for fc. The central value is taken as midway between these extremes.

Table 5.13: Factors contributing to uncertainties in the estimates of the first indirect forcing over Northern Hemisphere marine locations by aerosols associated with fossil fuels and other industrial processes and their estimated range.
Quantity	Central Value	2/3 Uncertainty Range
BackgroundNdfor Northern Hemisphere marine locations (cm-3)	140	66 to 214
PerturbedNdfor Northern Hemisphere marine location (cm-3)	217	124 to 310
Cloud mean liquid-water content (LWC) (g cm-3)	0.225	0.125 to 0.325
Background sulphate concentration (µg m-3)a	1.5	0.85 to 2.15
Cloud layer thickness (m)	200	100 to 300
Perturbed sulphate concentration (µg m-3)b	3.6	2.4 to 4.8
Susceptible cloud fraction, fc	0.24	0.19 to 0.29
Atmospheric transmission above cloud layer, Ta	0.92	0.78 to 1.00
Mean surface albedo	0.06	0.03 to 0.09
Result: If central value is –1.4 Wm–2 the 2/3 uncertainty range is from 0 to –2.8 Wm–2
a Calculated from a central value for the Northern Hemisphere marine burden of 0.12 TgS with an uncertainty range from 0.07 to 0.17 TgS. The central estimate for sulphur emissions in the Northern Hemisphere was 12 TgS/yr with an uncertainty range from 9 to 17 TgS/yr. b Calculated from a central value for the Northern Hemisphere marine perturbed burden of 0.28 TgS with an uncertainty range from 0.15 to 0.41 TgS. The central value for the Northern Hemisphere emissions of natural and anthropogenic sulphur was 74 TgS/yr with a 2/3 uncertainty range of 57 to 91 Tg S/yr.

The above assumptions, together with the parameter values given in Table 5.13, yield a central value for the indirect forcing over marine areas of -1.4 Wm^–2 together with an uncertainty of ±1.4 Wm^–2. Hence, the forcing lies in the range between zero and -2.8 Wm^–2. This range is in reasonable agreement with that given in Chapter 6, based on GCM assessments. Nearly all of the uncertainty in the forcing is associated with that in the planetary albedo which, in turn, is dominated by the uncertainties in the perturbed and unperturbed cloud albedos, and thus in the cloud optical depths. However, it is interesting to note that most of the uncertainty in the optical depths for both perturbed and unperturbed clouds arises from the uncertainties in the cloud LWC and cloud thickness, and not from the uncertainties in the CDNC. This is not actually particularly surprising and is simply due to the stronger functional dependence of the optical depth on the LWC and thickness, h. This allocation of uncertainty may be slightly misleading because the 1st indirect effect depends on an assumption of fixed h and LWC. Yet here we are concerned with evaluating both the central estimate as well as its uncertainty. Hence, knowledge of h and LWC are needed. Indeed, consideration of these parameters and the accuracy of their representation in GCMs must certainly contribute to any uncertainty in the estimates of forcing from GCMs. Thus, it seems clear that progress in reducing the uncertainty in the indirect forcing of the first kind will be at least as dependent on acquiring better data on cloud LWC and thickness as on better quantifying anthropogenic aerosol concentrations and their effects on N_d. At a somewhat lower priority, it is clearly quite important to better understand the relationship between cloud drop number concentration and sulphate concentration. The total uncertainty is clearly dependent on this relationship. If it were significantly different in form, a somewhat different order of contribution to total uncertainty might have arisen. On the other hand, the uncertainty in the magnitude of the emissions of sulphur gases as well as the uncertainty in the burden of sulphate played only a minor role in the determination of the overall uncertainty.

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