Abstract
I extend the reduced Greenland ice sheet (GIS) model-module of DICE-GIS (Nordhaus, Proc Natl Acad Sci 25(116):12261–12269, 2019) by integrating snow-albedo feedback (SAF) and potential tipping of the ice sheet into the resuming DICE-GIS SAF model. The increasing global temperature no longer only results in the melting of the GIS, but also in albedo loss, which in turn impacts the strength of the SAF. As a consequence, global warming and the melting of the ice accelerate. The social cost of carbon (SCC) increases because the economic damages are not only related to intensified sea level rise, but also to accelerated global temperature rise. Accounting for the SAF raises the SCC from 37 to 41 $/t CO2 in 2020, an increase of 11%. The temperature increase is the key channel through which the SAF impacts the SCC. The long-term volume of the GIS decreases by 2%, while the additional inclusion of tipping reduces it further by up to 35%.
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Notes
Area with positive mass balance, i.e., snow accumulation exceeds the melting of snow and ice discharge.
Area with negative mass balance, i.e., the melting of snow and ice discharge exceed the snow accumulation.
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Appendices
Appendix 1: Derivation of Eq. (8)
- Equation: :
-
$$ A(t) = \left\{\begin{array}{ll} A_{0} & \text{for} ~ V(t) \ge ~ \delta_{V} \\ A_{0} ~ \frac{V(t)}{\eta ~V(t) + (1-\eta)~\frac{\delta_{V}}{100} ~ V_{0}} & \text{for} ~ V(t) < ~\delta_{V}. \end{array}\right. $$
- Assumptions: :
-
-
1.
\(A(t) = \frac {V(t)}{H(t)}\)
-
2.
\(\dot {H}(t)= \eta _{1} ~ \dot {V}(t)\) for all t with V (t) ≥ δV, whereby η1 = 1/A0
-
3.
\(\dot {H}(t)=\eta _{2} ~ \dot {V}(t)\) for all t with V (t) < δV, whereby η2 = η/A0 and η ∈ (0, 1)
-
1.
- Case: :
-
V (t) ≥ δV
I assume that A(t) = A0, i.e., the surface area is the initial surface area and is constant over time. According to Assumption 2, the growth rate of H(t), denoted \(\dot H(t)\), is given as
with \(\dot V(t)\) denoting the growth rate of V (t). Integrating \(\dot H(t)\) gives
σ1 is chosen such that
Hence, σ1 = 0, and thus
Under Assumption 1, it follows for V (t) ≥ δV
- Case: :
-
V (t) < δV
According to Assumption 3, it is
Integrating \(\dot H(t)\) gives
σ2 is chosen such that H(t) is a continuous function with t∗ denoting the period when δV is reached. Hence, V (t∗) = δV. This gives
Therefore, it follows for σ2
For t ≥ t∗ it follows
Under Assumption 1, it follows for V (t) < δV
Appendix 2: Calibration tables
Appendix 3: Non-optimal policy results
The results for the non-optimal policy are presented in the following. The main difference to the results for the optimal policy is mentioned.
3.1 A3.1 SCC under non-optimal policy
Figure 5a shows the SCC until 2100 and Fig. 5b presents the peak values for the SCC from 2350 to 2500. The gray graph refers to DICE-GIS, the orange graph to DICE-GIS SAF, and the green graph to DICE-GIS SAF with multiple tipping.
The peak SCC for DICE-GIS is remarkably lower than under the optimal policy. When accounting for the SAF, the SCC is a little higher and the difference in the SCC between DICE-GIS SAF and DICE-GIS SAF with multiple tipping is larger.
3.2 A3.2 GIS dynamics
Figure 6 displays the volume of the GIS on the left axis (solid lines) and the surface fraction of the GIS on the right axis (dashed lines) from 2015 to 3500. Compared to the optimal policy, the volume decreases faster and is significantly lower in the long run. Consequently, the surface fraction starts to decrease earlier. Under the non-optimal policy, multiple tipping is reached 100 years in advance in 2280. Moreover, the shrinking of the surface area for DICE-GIS SAF with multiple tipping begins 180 years earlier in 2380. For DICE-GIS SAF, the surface fraction begins to decline in 2480.
Figure 7 presents the melt dynamics. In addition to the earlier passing of the tipping thresholds, the melt rate has a higher (more negative) magnitude, which leads to the greater volume loss (see Fig. 6). In contrast to the optimal policy, DICE-GIS SAF shows, in the long run, a slightly higher magnitude of the melt rate compared to DICE-GIS.
3.3 A3.3 Temperature dynamics
Figure 8 shows the temperature increase Tglobe on the left axis (solid lines) and the feedback parameter on the right axis (dashed lines). The peak increase in temperature is notably higher than under the optimal policy. When accounting for the SAF, the temperature increase shows only a very limited cooling after peaking. Hence, the temperature for DICE-GIS SAF remains at a relatively high level. The threshold κT is passed in 2085, 10 years earlier compared to the optimal policy. Surprisingly, the temperature increase is much higher for DICE-GIS SAF than for DICE-GIS SAF with multiple tipping.
In general, the feedback parameter λFB decreases more than with the optimal policy. After the surface of the GIS starts to decrease in 2380 for DICE-GIS SAF with multiple tipping and in 2480 for DICE-GIS SAF (see also Fig. 6), λFB continues to decline steadily. In contrast to the optimal policy, there is no subsequent increase in the feedback parameter.
Appendix 4: Sensitivity analysis
The following appendix presents a comprehensive sensitivity analysis for the parameters mentioned in Section 4. The calculations are based on DICE-GIS SAF. Several calculations include tipping. Volume tipping refers to the solely passing of κV. While temperature tipping refers to the solely passing of κT. Multiple tipping refers to the passing of both thresholds κV and κT.
4.1 A4.1 Volume and surface: η
The following analysis is based on DICE-GIS SAF with multiple tipping, because for DICE-GIS SAF, the volume does not fall below δV = 80. Figure 9 shows the SCC in dependence to η. Varying η in the range of 0.25 to 0.75 gives no significant change in the SCC in the year 2100. There is an absolute deviation in the peak SCC of − 28.20 to 18.55 $/t CO2. This gives a relative deviation of − 0.65 to 0.40%.
Figure 10 displays the volume in dependence to η. The volume in 3500 gives an absolute deviation from − 0.94 to 1.40%p (percentage points).
The higher η, the more convex is the relation between the loss of the volume and the decrease of the surface fraction. Hence, the higher η, the higher is the surface fraction for a given volume. This implies less total albedo loss and thus a lower decrease in the feedback parameter after passing δV. Therefore, the temperature increase is negatively correlated to η. Hence, the SCC is relatively lower while the volume is relatively higher.
I conclude that the SCC and the volume are relatively robust to the variation of η.
4.2 A4.2 Volume and surface: δ V
The following analysis is based on DICE-GIS SAF with volume tipping. There is no significant deviation in the SCC when varying δV from 76 to 85. The absolute deviation of the peak SCC is between − 0.2 to 0.7 $/tCO2.
Figure 11 shows the impact on the volume. The absolute deviation in 3500 is in the range of − 0.26 to 0.09%p.
Note, that the volume starts to deviate earlier for higher values of δV, because as the surface fraction declines, there is total albedo loss and the feedback parameter decreases again (see Fig. 4b). This, in turn, accelerates the temperature increase and promotes the melting of the GIS.
In addition, I vary δV from 99.9 to 91 to investigate the impact on DICE-GIS SAF as well. For δV = 99.9, the surface fraction starts to decrease almost immediately as the volume melts.
Figure 12 presents the SCC depending on δV. The SCC in 2100 shows an absolute deviation of 2.29 $/t CO2, or relatively 0.7%. The peak SCC deviates up to 131.63 $/t CO2, or 3.1%.
Figure 13 shows the volume in dependence to δV. The GIS starts to melt comparatively earlier for higher values of δV, which reveals a similar pattern as in Fig. 11. The volume in 3500 decreases by up to − 0.83%p.
Consequently, the volume is relatively robust to the variation of δV. This holds for DICE-GIS SAF as well as for DICE-GIS SAF with volume tipping. The SCC is sensitive for DICE-GIS SAF under δV ≥ 91, but not for volume tipping under 75 ≤ δV ≤ 85.
4.3 A4.3 SAF: λ SAF
The following analysis is based on DICE-GIS SAF. Figure 14 presents the SCC in dependence to λSAF in the range of 0.25 to 0.75 (Flanner et al. 2011; Duan et al. 2019; Qu and Hall 2014; Winton 2006). The SCC in 2100 differs between − 23.83 and 27.47 $/tCO2, or − 7.5 to 8.5%. The deviation of the peak SCC is − 359.90 to 431.51 $/tCO2, or − 8.5 to 10.2%.
Figure 15 shows the influence of the absolute contribution of the SAF to the climate feedback parameter λSAF. The higher λSAF, the stronger is the decreasing impact on the global feedback parameter λFB. Thereby, the course remains almost robust. The feedback parameter in 3500 is in the range of 1.06 to 1.15, a relative deviation of − 4.6 to 3.6%.
I deduce that the SCC and the feedback parameter are sensitive to λSAF. This underlines the fact that the SAF plays an important role for the climate and has a significant impact on the SCC.
4.4 A4.4 SAF: γ
The following analysis is based on DICE-GIS SAF. Figure 16 displays the SCC in dependence to γ. The SCC in 2100 varies between − 5.89 and 6.09 $/tCO2, or − 1.8 to 1.9%. The peak SCC deviates from − 89.77 to 93.77 $/tCO2, or − 2.1 to 2.2%.
I conclude that the SCC is sensitive to γ, which underlines the impact of the SAF.
4.5 A4.5 Tipping: κ V
The following analysis is based on DICE-GIS SAF with volume tipping. Figure 17 visualizes the SCC with κV varying from 95 to 85. There is no significant impact on the SCC in 2100. The peak SCC deviates in the range of − 2.96 to 22.65 $/tCO2, or − 0.06 to 0.5%. The volume deviates in the range of − 0.05 to 0.06%p.
I derive a limited sensitivity of the SCC to κV, while the volume is robust. This is because κV is reached very late (see Fig. 2a).
4.6 A4.6 Tipping: κ T
The following analysis is based on DICE-GIS SAF with temperature tipping. Figure 18 presents the SCC with κT varying from 1.5 to 4.5 °C (IPCC 2019; Robinson et al. 2012; Ridley et al. 2009). The SCC in 2100 falls by up to − 1.52 $/t CO2, or 0.04%. The peak SCC deviates in the range of − 20.64 to 0.50 $/tCO2, or − 0.5 to 0.01%.
Figure 19 shows the volume depending on κT and depicts a similar pattern. The volume is only sensitive for κT ≥ 3.6 °C and deviates up to 13.0%p.
Since the thresholds for κT < 3.6 °C are all reached very early (see Fig. 4b), temperature tipping starts at about the same time for 1.5 ≤ κT < 3.6. Therefore, neither the SCC nor the volume shows a significant deviation. For κT ≥ 3.6 °C, the SCC and the volume are sensitive as the timing of passing the threshold varies.
4.7 A4.7 Tipping: ρ
The following analysis is based on DICE-GIS SAF with multiple tipping. Figure 20 shows the SCC in dependence to ρ varying from 1.5 to 2.5. There is no significant variation in the SCC in 2100. The peak SCC deviates in the range of − 50.23 to 89.60 $/tCO2, or by − 1.2 to 2.1%.
The volume deviates in the range of − 0.19 to 0.18%p. I deduce that the SCC and particularly the volume are rather robust to ρ.
4.8 A4.8 Tipping: ψ
The following analysis incorporates DICE-GIS SAF with volume, temperature and multiple tipping. ψ is varied in the range of 1.5 to 4. Figure 21 presents the SCC in variation to ψ.
For DICE-GIS SAF with volume tipping, there is no deviation in 2100. But, the peak SCC varies in the range of − 1.37 to 6.66 $/tCO2, or − 0.03 to 0.16%. Surprisingly the SCC for ψ = 1.75 initially shows a positive deviation followed by a negative deviation. For DICE-GIS SAF with temperature tipping, the SCC in 2100 varies from − 0.75 to 2.95 $/tCO2, or − 0.02 to 0.09%. The deviation of the peak SCC is between − 10.63 to 171.29 $/tCO2, or − 0.2 to 4.0%. For DICE-GIS SAF with multiple tipping and ψ ≤ 3, the SCC in 2100 varies from − 0.79 to 2.27 $/tCO2, or − 0.2 to 0.7%. The deviation of the peak SCC is − 73.94 to 467.10 $/tCO2, or − 1.7 to 10.8%. The SCC is highly sensitive for ψ > 3, exploding up to 3.9 million $/tCO2.
Figure 22 shows the volume in variation to ψ. Accounting for volume tipping, the volume in 3500 is in the range of 62.16 to 77.54, or − 11.92 to 3.46%p. Accounting for temperature tipping, it is in the range of 48.99 to 74.39, or − 19.27 to 6.12%p. Accounting for multiple tipping, it is in the range of 19.06 to 68.13, or − 34.02 to 15.05%p.
I conclude that, surprisingly, there is only a very small impact for volume tipping. Temperature tipping is quite sensitive to ψ. While accounting for multiple tipping is the most sensitive, because ψ is exponentiated with ρ (see Eq. (15)).
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Gschnaller, S. The albedo loss from the melting of the Greenland ice sheet and the social cost of carbon. Climatic Change 163, 2201–2231 (2020). https://doi.org/10.1007/s10584-020-02936-7
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DOI: https://doi.org/10.1007/s10584-020-02936-7