# 10.A.2 Mass Balance Sensitivity of Glaciers and Ice Caps

A linear relationship r_{g} = b_{g} × (T − T_{0}) is found for the period 1961 to 2003 between the observational time series of the contribution r_{g} to the rate of sea level rise from the world’s glaciers and ice caps (G&IC, excluding those on Antarctica and Greenland; Section 4.5.2, Figure 4.14) and global average surface air temperature T (Hadley Centre/Climatic Research Unit gridded surface temperature dataset HadCRUT3; Section 3.2.2.4, Figure 3.6), where b_{g} is the global total G&IC mass balance sensitivity and T_{0} is the global average temperature of the climate in which G&IC are in a steady state, T and T_{0} being expressed relative to the average of 1865 to 1894. The correlation coefficient is 0.88. Weighted least-squares regression gives a slope b_{g} = 0.84 ± 0.15 (one standard deviation) mm yr^{–1} °C^{–1}, with T_{0} = −0.13°C. Surface mass balance models driven with climate change scenarios from AOGCMs (Section 10.6.3.1) also indicate such a linear relationship, but the model results give a somewhat lower b_{g} of around 0.5 to 0.6 mm yr^{–1} °C^{–1} (Section 10.6.3.1). To cover both observations and models, we adopt a value of b_{g} = 0.8 ± 0.2 (one standard deviation) mm yr^{–1} °C^{–1}. This uncertainty of ±25% is smaller than that of ±40% used in the TAR because of the improved observational constraint now available. To make projections, we choose a set of values of b_{g} randomly from a normal distribution. We use T_{0} = T - rg/bg , where T = 0.40 °C and rg = 0.45 mm yr^{–1}, are the averages over the period 1961 to 2003. This choice of T_{0} minimises the root mean square difference of the predicted r_{g} from the observed, and gives T_{0} in the range −0.5°C to 0.0°C (5 to 95%). Note that a constant b_{g} is not expected to be a good approximation if glacier area changes substantially (see Section 10.A.3).

## 10.A.3 Area Scaling of Glaciers and Ice Caps

Model results using area-volume scaling of G&IC (Section 10.6.3.2) are approximately described by the relations b_{g} / b_{1} = (A_{g} / A_{1})^{1.96} and A_{g} / A_{1} = (V_{g} / V_{1})^{0.84}, where A_{g} and V_{g} are the global G&IC area and volume (excluding those on Greenland and Antarctica) and variable X_{1} is the initial value of X_{g}. The first relation describes how total SMB sensitivity declines as the most sensitive areas are ablated most rapidly. The second relation follows Wigley and Raper (2005) in its form, and describes how area declines as volume is lost, with dV_{g} / dt = −r_{g} (expressing V as sea level equivalent, i.e., the liquid-water-equivalent volume of ice divided by the surface area of the world ocean). Projections are made starting from 1990 using T from Section 10.A.1 with initial values of the present-day b_{g} from Section 10.A.2 and the three recent estimates V_{g} = 0.15, 0.24 and 0.37 m from Table 4.4, which are assumed equally likely. We use T = 0.48°C at 1990 relative to 1865 to 1894, and choose T_{0} as in Section 10.A.2. An uncertainty of 10% (one standard deviation) is assumed because of the scaling relations. The results are multiplied by 1.2 (Section 10.6.3.3) to include contributions from G&IC on Greenland and Antarctica (apart from the ice sheets). These scaling relations are expected to give a decreasingly adequate approximation as greater area and volume is lost, because they do not model hypsometry explicitly; they predict that V will tend eventually to zero in any steady-state warmer climate, for instance, although this is not necessarily the case. A similar scaling procedure was used in the TAR. Current estimates of present-day G&IC mass are smaller than those used in the TAR, leading to more rapid wastage of area. Hence, the central estimates for the G&IC contribution to sea level rise in Table 10.7 are similar to those in the TAR, despite our use of a larger mass balance sensitivity (Section 10.A.2).